Pythagorean Theorem Math Worksheet 23 Answer Key showing three problems with geometric representations and solutions.

Let’s solve each triangle one by one using the Pythagorean Theorem. Remember, for a right triangle with legs a and b, and hypotenuse c, the theorem says:

> a² + b² = c²

But in these problems, we’re given the areas of squares built on each side — which are just a², b², and c². So we can take square roots to find the actual side lengths.

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First Triangle (Top)

We’re told:
- Square on side a has area 9 → so a² = 9
- Square on side b has area 16 → so b² = 16
- Square on side c has area 25 → so c² = 25

Now take square roots:

- a = √9 = 3
- b = √16 = 4
- c = √25 = 5

Check: Does 3² + 4² = 5²?
→ 9 + 16 = 25 → YES! ✔

So sides are:
side a = 3
side b = 4
side c = 5

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Second Triangle (Middle)

Given:
- Square on side a has area 81 → a² = 81
- Square on side b has area 144 → b² = 144
- Square on side c has area 225 → c² = 225

Take square roots:

- a = √81 = 9
- b = √144 = 12
- c = √225 = 15

Check: 9² + 12² = 81 + 144 = 225 = 15² → YES! ✔

Sides:
side a = 9
side b = 12
side c = 15

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Third Triangle (Bottom)

Given:
- Square on side a has area 64 → a² = 64
- Square on side b has area 225 → b² = 225
- Square on side c has area 289 → c² = 289

Take square roots:

- a = √64 = 8
- b = √225 = 15
- c = √289 = 17

Check: 8² + 15² = 64 + 225 = 289 = 17² → YES! ✔

Sides:
side a = 8
side b = 15
side c = 17

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Final Answer:
First triangle: side a = 3, side b = 4, side c = 5
Second triangle: side a = 9, side b = 12, side c = 15
Third triangle: side a = 8, side b = 15, side c = 17