EASYTUTORGUIDE

Practical tutorials, tools, courses, digital skills, and business promotion.

Free Learning
Google Translate — English / فارسی / العربية
EasyTutorGuide LogoLearn, Build, & Grow

Chapter 38: Pythagorean Theorem

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
Advertisement area — AdSense / Auto Ads

What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

38.1 Right triangles

An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Right triangles.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A triangle has angles 35° and 45°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-35-45=100.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 100°

Worked Example 2

Problem: A triangle has angles 48° and 58°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-48-58=74.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 74°

Worked Example 3

Problem: A triangle has angles 67° and 47°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-67-47=66.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 66°

Worked Example 4

Problem: A triangle has angles 72° and 52°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-72-52=56.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 56°

Worked Example 5

Problem: A triangle has angles 110° and 60°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-110-60=10.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 10°

Worked Example 6

Problem: A triangle has angles 25° and 65°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-25-65=90.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 90°

Worked Example 7

Problem: A triangle has angles 58° and 68°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-58-68=54.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 54°

Worked Example 8

Problem: A triangle has angles 83° and 63°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-83-63=34.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 34°

Worked Example 9

Problem: A triangle has angles 95° and 45°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-95-45=40.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 40°

Worked Example 10

Problem: A triangle has angles 120° and 40°. Find the third angle.

  1. Triangle interior angles total 180°.
  2. 180-120-40=20.

Very beginner explanation: Subtract the known angles from 180°.

Answer: 20°

Practice exercise

Create one new Grade 8 problem involving Right triangles. Show the important steps, include units when needed, and explain how you checked the answer.

38.2 Legs

Legs is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Legs: Explain Legs in one simple sentence.

  1. Look at the words in “Legs”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Legs is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Legs: A student says, “I can use Legs without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Legs: What is the first step when solving a problem about Legs?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Legs: After solving a Legs problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Legs: Give one way Legs could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Legs can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Legs: Which representation could help explain Legs: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Legs: A student gets an answer for Legs but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Legs: Why can estimation help before a detailed Legs calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Legs: How can you test whether your rule for Legs works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Legs: How would you teach Legs to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Legs. Show the important steps, include units when needed, and explain how you checked the answer.

38.3 Hypotenuse

The hypotenuse is the longest side of a right triangle and is opposite the right angle. In this section, the focus is Hypotenuse.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

Create one new Grade 8 problem involving Hypotenuse. Show the important steps, include units when needed, and explain how you checked the answer.

38.4 Meaning of a² + b² = c²

The mean is the sum of all values divided by the number of values. In this section, the focus is Meaning of a² + b² = c².

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the mean of [4, 6, 8].

  1. Add the values: 18.
  2. Divide by 3.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 6

Worked Example 2

Problem: Find the mean of [5, 9, 10, 12].

  1. Add the values: 36.
  2. Divide by 4.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 9

Worked Example 3

Problem: Find the mean of [3, 7, 7, 11].

  1. Add the values: 28.
  2. Divide by 4.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 7

Worked Example 4

Problem: Find the mean of [20, 25, 30].

  1. Add the values: 75.
  2. Divide by 3.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 25

Worked Example 5

Problem: Find the mean of [6, 8, 9, 12, 15].

  1. Add the values: 50.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 10

Worked Example 6

Problem: Find the mean of [2, 4, 4, 5, 20].

  1. Add the values: 35.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 7

Worked Example 7

Problem: Find the mean of [10, 11, 12, 13, 14].

  1. Add the values: 60.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 12

Worked Example 8

Problem: Find the mean of [1, 3, 5, 7, 9].

  1. Add the values: 25.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 5

Worked Example 9

Problem: Find the mean of [8, 8, 8, 9, 10].

  1. Add the values: 43.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 8.6

Worked Example 10

Problem: Find the mean of [15, 18, 21, 24, 27].

  1. Add the values: 105.
  2. Divide by 5.

Very beginner explanation: The mean shares the total equally among all values.

Answer: 21

Practice exercise

Create one new Grade 8 problem involving Meaning of a² + b² = c². Show the important steps, include units when needed, and explain how you checked the answer.

38.5 Finding the hypotenuse

The hypotenuse is the longest side of a right triangle and is opposite the right angle. In this section, the focus is Finding the hypotenuse.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

Create one new Grade 8 problem involving Finding the hypotenuse. Show the important steps, include units when needed, and explain how you checked the answer.

38.6 Finding a missing leg

Finding a missing leg is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has hypotenuse 5 and one leg 3. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 5² - 3² = 16.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 4

Worked Example 2

Problem: A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 13² - 5² = 144.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 12

Worked Example 3

Problem: A right triangle has hypotenuse 10 and one leg 6. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 10² - 6² = 64.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 8

Worked Example 4

Problem: A right triangle has hypotenuse 17 and one leg 8. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 17² - 8² = 225.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 15

Worked Example 5

Problem: A right triangle has hypotenuse 25 and one leg 7. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 25² - 7² = 576.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 24

Worked Example 6

Problem: A right triangle has hypotenuse 15 and one leg 9. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 15² - 9² = 144.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 12

Worked Example 7

Problem: A right triangle has hypotenuse 20 and one leg 12. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 20² - 12² = 256.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 16

Worked Example 8

Problem: A right triangle has hypotenuse 29 and one leg 20. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 29² - 20² = 441.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 21

Worked Example 9

Problem: A right triangle has hypotenuse 26 and one leg 10. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 26² - 10² = 576.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 24

Worked Example 10

Problem: A right triangle has hypotenuse 25 and one leg 15. Find the other leg.

  1. Use a² + b² = c².
  2. b² = 25² - 15² = 400.
  3. Take the positive square root.

Very beginner explanation: The hypotenuse is always the side opposite the right angle and is the longest side.

Answer: 20

Practice exercise

Create one new Grade 8 problem involving Finding a missing leg. Show the important steps, include units when needed, and explain how you checked the answer.

38.7 Square roots in Pythagorean problems

A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots in Pythagorean problems.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find √4.

  1. Ask which positive number multiplied by itself equals 4.
  2. 2 × 2 = 4.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 2

Worked Example 2

Problem: Find √9.

  1. Ask which positive number multiplied by itself equals 9.
  2. 3 × 3 = 9.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 3

Worked Example 3

Problem: Find √16.

  1. Ask which positive number multiplied by itself equals 16.
  2. 4 × 4 = 16.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 4

Worked Example 4

Problem: Find √25.

  1. Ask which positive number multiplied by itself equals 25.
  2. 5 × 5 = 25.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 5

Worked Example 5

Problem: Find √36.

  1. Ask which positive number multiplied by itself equals 36.
  2. 6 × 6 = 36.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 6

Worked Example 6

Problem: Find √49.

  1. Ask which positive number multiplied by itself equals 49.
  2. 7 × 7 = 49.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 7

Worked Example 7

Problem: Find √64.

  1. Ask which positive number multiplied by itself equals 64.
  2. 8 × 8 = 64.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 8

Worked Example 8

Problem: Find √81.

  1. Ask which positive number multiplied by itself equals 81.
  2. 9 × 9 = 81.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 9

Worked Example 9

Problem: Find √100.

  1. Ask which positive number multiplied by itself equals 100.
  2. 10 × 10 = 100.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 10

Worked Example 10

Problem: Find √144.

  1. Ask which positive number multiplied by itself equals 144.
  2. 12 × 12 = 144.

Very beginner explanation: Square root reverses the operation of squaring.

Answer: 12

Practice exercise

Create one new Grade 8 problem involving Square roots in Pythagorean problems. Show the important steps, include units when needed, and explain how you checked the answer.

38.8 Determining whether a triangle is right

A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Determining whether a triangle is right.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

Create one new Grade 8 problem involving Determining whether a triangle is right. Show the important steps, include units when needed, and explain how you checked the answer.

38.9 Distance applications

Distance applications is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

Create one new Grade 8 problem involving Distance applications. Show the important steps, include units when needed, and explain how you checked the answer.

38.10 Diagonal applications

Diagonal applications is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

Create one new Grade 8 problem involving Diagonal applications. Show the important steps, include units when needed, and explain how you checked the answer.

38.11 Coordinate-grid applications

Coordinates locate a point using an ordered pair (x, y), with horizontal movement first and vertical movement second. In this section, the focus is Coordinate-grid applications.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

Create one new Grade 8 problem involving Coordinate-grid applications. Show the important steps, include units when needed, and explain how you checked the answer.

38.12 Real-life Pythagorean problems

The Pythagorean theorem relates the side lengths of a right triangle: a² + b² = c², where c is the hypotenuse. In this section, the focus is Real-life Pythagorean problems.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A right triangle has legs 3 and 4. Find the hypotenuse.

  1. c² = 3² + 4².
  2. c² = 25.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 5

Worked Example 2

Problem: A right triangle has legs 5 and 12. Find the hypotenuse.

  1. c² = 5² + 12².
  2. c² = 169.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 13

Worked Example 3

Problem: A right triangle has legs 6 and 8. Find the hypotenuse.

  1. c² = 6² + 8².
  2. c² = 100.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 10

Worked Example 4

Problem: A right triangle has legs 8 and 15. Find the hypotenuse.

  1. c² = 8² + 15².
  2. c² = 289.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 17

Worked Example 5

Problem: A right triangle has legs 7 and 24. Find the hypotenuse.

  1. c² = 7² + 24².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Worked Example 6

Problem: A right triangle has legs 9 and 12. Find the hypotenuse.

  1. c² = 9² + 12².
  2. c² = 225.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 15

Worked Example 7

Problem: A right triangle has legs 12 and 16. Find the hypotenuse.

  1. c² = 12² + 16².
  2. c² = 400.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 20

Worked Example 8

Problem: A right triangle has legs 20 and 21. Find the hypotenuse.

  1. c² = 20² + 21².
  2. c² = 841.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 29

Worked Example 9

Problem: A right triangle has legs 10 and 24. Find the hypotenuse.

  1. c² = 10² + 24².
  2. c² = 676.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 26

Worked Example 10

Problem: A right triangle has legs 15 and 20. Find the hypotenuse.

  1. c² = 15² + 20².
  2. c² = 625.
  3. Take the positive square root.

Very beginner explanation: The Pythagorean theorem applies only to right triangles.

Answer: 25

Practice exercise

Create one new Grade 8 problem involving Real-life Pythagorean problems. Show the important steps, include units when needed, and explain how you checked the answer.

38.13 Multi-step problems

Multi-step problems is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Multi-step problems: Explain Multi-step problems in one simple sentence.

  1. Look at the words in “Multi-step problems”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Multi-step problems is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Multi-step problems: A student says, “I can use Multi-step problems without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Multi-step problems: What is the first step when solving a problem about Multi-step problems?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Multi-step problems: After solving a Multi-step problems problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Multi-step problems: Give one way Multi-step problems could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Multi-step problems can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Multi-step problems: Which representation could help explain Multi-step problems: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Multi-step problems: A student gets an answer for Multi-step problems but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Multi-step problems: Why can estimation help before a detailed Multi-step problems calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Multi-step problems: How can you test whether your rule for Multi-step problems works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Multi-step problems: How would you teach Multi-step problems to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

Create one new Grade 8 problem involving Multi-step problems. Show the important steps, include units when needed, and explain how you checked the answer.

Advertisement area — AdSense / Auto Ads

Chapter 38 Review Questions and Answers

Q1. What is important to remember about Right triangles?

Answer: An angle is formed by two rays meeting at a vertex and is measured in degrees. In this section, the focus is Right triangles.

Q2. What is important to remember about Legs?

Answer: Legs is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Hypotenuse?

Answer: The hypotenuse is the longest side of a right triangle and is opposite the right angle. In this section, the focus is Hypotenuse.

Q4. What is important to remember about Meaning of a² + b² = c²?

Answer: The mean is the sum of all values divided by the number of values. In this section, the focus is Meaning of a² + b² = c².

Q5. What is important to remember about Finding the hypotenuse?

Answer: The hypotenuse is the longest side of a right triangle and is opposite the right angle. In this section, the focus is Finding the hypotenuse.

Q6. What is important to remember about Finding a missing leg?

Answer: Finding a missing leg is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Square roots in Pythagorean problems?

Answer: A square root (a value that multiplied by itself gives the original number) reverses squaring. In this section, the focus is Square roots in Pythagorean problems.

Q8. What is important to remember about Determining whether a triangle is right?

Answer: A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Determining whether a triangle is right.

Q9. What is important to remember about Distance applications?

Answer: Distance applications is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q10. What is important to remember about Diagonal applications?

Answer: Diagonal applications is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q11. What is important to remember about Coordinate-grid applications?

Answer: Coordinates locate a point using an ordered pair (x, y), with horizontal movement first and vertical movement second. In this section, the focus is Coordinate-grid applications.

Q12. What is important to remember about Real-life Pythagorean problems?

Answer: The Pythagorean theorem relates the side lengths of a right triangle: a² + b² = c², where c is the hypotenuse. In this section, the focus is Real-life Pythagorean problems.

Q13. What is important to remember about Multi-step problems?

Answer: Multi-step problems is an important Grade 8 concept in Pythagorean Theorem. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.