Pythagorean 2D

Grade 8 · trigonometry · 77 practice problems · read aloud

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What is the Pythagorean Theorem? 🔺

The Pythagorean Theorem is a rule that works for all right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is super 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.

The famous formula is: a² + b² = c², where a and b are the legs, and c is the hypotenuse.

How to Solve a Problem: Step-by-Step

  1. Identify the sides. Find the legs (a, b) and the hypotenuse (c). The hypotenuse is always the longest side, across from the right angle.
  2. Plug into the formula. Write down a² + b² = c².
  3. Substitute the known values. Replace the letters with the numbers you know.
  4. Solve for the unknown. Perform the calculations step-by-step to find the missing side.
  5. Don't forget the square root! If you solved for c², take the square root to find c.

Worked Examples

Example 1: Find the Hypotenuse

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

  1. Formula: a² + b² = c²
  2. Substitute: 6² + 8² = c²
  3. Calculate: 36 + 64 = c² → 100 = c²
  4. Square Root: √100 = c → c = 10 cm

Example 2: Find a Leg

The hypotenuse of a right triangle is 13 m, and one leg is 5 m. Find the other leg (a).

  1. Formula: a² + b² = c²
  2. Substitute: a² + 5² = 13²
  3. Calculate: a² + 25 = 169
  4. Isolate a²: a² = 169 - 25 → a² = 144
  5. Square Root: √144 = a → a = 12 m

Common Mistakes to Avoid

  • Using the theorem on non-right triangles. Remember, it only works if there is a 90° angle.
  • Forgetting to take the square root. If you end with c² = 25, then c = 5!
  • Mixing up the hypotenuse and a leg. The hypotenuse (c) is always the longest side. If you're solving for a leg, you must subtract.
  • Adding sides before squaring. You must square each side first, then add them. (3+4)² is not the same as 3²+4²!

Tips & Tricks

  • Remember the formula with this phrase: "The square of the hypotenuse is equal to the sum of the squares of the other two sides."
  • Look for common "Pythagorean Triples" like 3-4-5, 5-12-13, and 8-15-17. If you see sides that match these ratios, you can solve instantly!
  • Always double-check that your answer makes sense. The hypotenuse must be longer than either leg.

How to Practice

To get good at this, try these activities:

  • Solve at least 5 practice problems every day.
  • Draw the triangle for every problem—it helps you visualize the parts.
  • Create your own real-world problems (e.g., "How long is the kite string?") and solve them.
  • Use online quizzes and flashcards to memorize the common Pythagorean triples.

Practice problems

6 of the 77, worked through step by step — try them before opening the answer.

1 √(6² + 8²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Remember the relationship between the sides of a right triangle.

Show the answer

Answer: 10

  1. Calculate 6 squared** 6² = 6 × 6 = 36 **
  2. Calculate 8 squared** 8² = 8 × 8 = 64 **
  3. Add the results** 36 + 64 = 100 **
  4. Take the square root** √100 = 10 **Final Answer:** 10

Let's solve step by step. We are given: √(6² + 8²) = ? **

2 √(15² + 8²) = ?

Hint: This involves finding the length of the longest side in a right triangle when you know the lengths of the two shorter sides. Remember the relationship between these three sides.

Show the answer

Answer: 17

  1. Calculate the squares** 15² = 15 × 15 = 225 8² = 8 × 8 = 64 **
  2. Add the squares** 225 + 64 = 289 **
  3. Take the square root** √289 = ? We know 17 × 17 = 289, so √289 = 17. **Final Answer:** 17

Let's solve step-by-step. We are given: √(15² + 8²) **

3 √(9² + 12²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Remember the relationship between the sides of a right triangle.

Show the answer

Answer: 15

  1. Square both numbers: 9² = 81 and 12² = 144
  2. Add the squared values: 81 + 144 = 225
  3. Take the square root of the sum: √225 = 15

The answer is 15.

4 √(13² - 5²) = ?

Hint: This involves finding the length of one side of a right triangle when you know the hypotenuse and another side. Remember the relationship between the sides in the Pythagorean theorem.

Show the answer

Answer: 12

  1. Write the Pythagorean theorem: a² + b² = c²
  2. We have c = 13 and b = 5, so we need to find a: a² + 5² = 13²
  3. Calculate the squares: a² + 25 = 169
  4. Subtract 25 from both sides: a² = 169 - 25
  5. Simplify: a² = 144
  6. Take the square root: a = √144
  7. √144 = 12

The answer is 12.

5 √(12² + 16²) = ?

Hint: This involves finding the length of the longest side in a right triangle when you know the other two sides. Remember the relationship between the sides in such triangles.

Show the answer

Answer: 20

  1. Calculate the squares inside the square root. 12² = 12 × 12 = 144 16² = 16 × 16 = 256
  2. Add the two results. 144 + 256 = 400
  3. Take the square root of the sum. √400 = 20 So the final answer is 20. Explanation: This expression is the Pythagorean formula for the hypotenuse of a right triangle with legs 12 and 16. The calculation shows that the hypotenuse length is 20.

We are given: √(12² + 16²)

6 √(15² + 20²) = ?

Hint: This expression involves finding the square root of the sum of two squared numbers. Think about the relationship between the three sides of a right triangle.

Show the answer

Answer: 25

  1. Calculate 15 squared** 15 × 15 = 225 **
  2. Calculate 20 squared** 20 × 20 = 400 **
  3. Add the two results** 225 + 400 = 625 **
  4. Take the square root** √625 = ? We know 25 × 25 = 625, so √625 = 25. **
  5. Final answer** The result is 25. **Reasoning:** This is an application of the Pythagorean theorem formula for the hypotenuse: c = √(a² + b²) Here a = 15, b = 20, so c = √(225 + 400) = √625 = 25.

Let's solve step by step. We are given: √(15² + 20²) **

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