Pythagorean Theorem

Grade 8 · trigonometry · 81 practice problems · read aloud

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The Pythagorean Theorem: Your Key to Right Triangles 🔑

1. What is it and Why is it Useful?

The Pythagorean Theorem is a rule that only works for right triangles (triangles with a 90° angle). It states that the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides (called legs).

Formula: a² + b² = c²

Where 'c' is always the hypotenuse. It's incredibly useful for finding missing side lengths in real-world problems, like figuring out how long a ladder needs to be or the shortest distance between two points.

2. Step-by-Step Guide

  1. Identify the right triangle. Make sure it has a 90° angle.
  2. Label the sides. Find the hypotenuse (longest side, opposite the right angle) and label it 'c'. Label the other two sides 'a' and 'b'.
  3. Plug into the formula. a² + b² = c²
  4. Solve for the unknown side. Substitute the known values and perform the calculations.
  5. Don't forget to square root! If you solved for a², b², or c², take the square root to find the final side length.

3. Visual Examples

Example 1: Find the Hypotenuse

A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse (c).

  1. a = 6, b = 8, c = ?
  2. a² + b² = c²
  3. 6² + 8² = c²
  4. 36 + 64 = c²
  5. 100 = c²
  6. √100 = c → c = 10 cm

Example 2: Find a Leg

The hypotenuse is 15 m, and one leg is 9 m. Find the other leg (a).

  1. a = ?, b = 9, c = 15
  2. a² + b² = c²
  3. a² + 9² = 15²
  4. a² + 81 = 225
  5. a² = 225 - 81
  6. a² = 144
  7. √144 = a → a = 12 m

4. Common Mistakes to Avoid

  • Using the theorem on non-right triangles. It only applies to triangles with a 90° angle.
  • Forgetting to identify the hypotenuse correctly. 'c' is ALWAYS the longest side.
  • Forgetting to take the square root at the end. If you get c² = 25, you're not done! c = 5.
  • Adding the sides before squaring them. a² + b² is NOT the same as (a + b)².

5. Tips & Tricks

  • 🔍 Look for common "Pythagorean Triples"—whole number side lengths like 3-4-5, 5-12-13, and 8-15-17. If you see two sides from one of these sets, you know the third instantly!
  • Remember the formula with a silly sentence: "The square of the hypotenuse is equal to the sum of the squares of the other two sides."
  • When solving for a leg, remember to subtract before you square root.

6. How to Practice

  • Draw it! For every problem, sketch the triangle and label the sides.
  • Start with basic problems to master the steps before moving to word problems.
  • Practice identifying Pythagorean Triples.
  • Look for real-life applications: measure the diagonal of your TV or the distance across a rectangular park.

Practice problems

6 of the 81, worked through step by step — try them before opening the answer.

1 (3² + 4²) = ?

Hint: Remember to calculate each exponent first before adding the results together.

Show the answer

Answer: 25

  1. Identify the operations inside the parentheses. We have (3² + 4²). This means we need to calculate 3 squared and 4 squared first.
  2. Calculate 3². 3² = 3 × 3 = 9.
  3. Calculate 4². 4² = 4 × 4 = 16.
  4. Add the results. 9 + 16 = 25.
  5. Final answer. (3² + 4²) = 25.

Let's solve the problem step by step.

2 √(7² + 24²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship connects these three values.

Show the answer

Answer: 25

  1. Calculate 7 squared: 7² = 49
  2. Calculate 24 squared: 24² = 576
  3. Add the two squared values: 49 + 576 = 625
  4. Take the square root of the sum: √625 = 25

The answer is 25.

3 √(8² + 15²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship connects these three values.

Show the answer

Answer: 17

  1. Calculate 8 squared: 8² = 64
  2. Calculate 15 squared: 15² = 225
  3. Add the two squared values: 64 + 225 = 289
  4. Take the square root of the sum: √289 = 17

The answer is 17.

4 √(9² + 40²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship connects these three values.

Show the answer

Answer: 41

  1. Calculate 9 squared: 9² = 81
  2. Calculate 40 squared: 40² = 1600
  3. Add the two squared values: 81 + 1600 = 1681
  4. Take the square root of the sum: √1681 = 41

The answer is 41.

5 √(5² + 12²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship connects these three values.

Show the answer

Answer: 13

  1. Calculate 5 squared: 5² = 25
  2. Calculate 12 squared: 12² = 144
  3. Add the two squared values: 25 + 144 = 169
  4. Take the square root of the sum: √169 = 13

The answer is 13.

6 √(12² + 35²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship connects these three values.

Show the answer

Answer: 37

  1. Calculate 12 squared: 12² = 144
  2. Calculate 35 squared: 35² = 1225
  3. Add the two squared values: 144 + 1225 = 1369
  4. Take the square root of the sum: √1369 = 37

The answer is 37.

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