Since I can't view or access images directly, I can't see the specific details of your worksheet. However, I can help you solve
Pythagorean Theorem problems based on typical setups like the ones described in your worksheet.
Let me walk you through how to solve these types of problems and provide a general solution approach. You can apply this method to each triangle on your worksheet.
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🔷 Pythagorean Theorem:
For a right triangle with legs $ a $ and $ b $, and hypotenuse $ c $:
$$
a^2 + b^2 = c^2
$$
If the side lengths form a
Pythagorean triple, they are whole numbers that satisfy this equation (e.g., 3-4-5, 5-12-13, etc.).
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✔ Steps to Solve Each Problem:
1.
Identify the sides: Label the two known sides as $ a $ and $ b $, and the unknown as $ c $ (hypotenuse) or one leg.
2.
Plug into the formula:
- If solving for the hypotenuse: $ c = \sqrt{a^2 + b^2} $
- If solving for a leg: $ a = \sqrt{c^2 - b^2} $
3.
Simplify the radical if needed.
4.
Check if it's a Pythagorean triple: Are all sides whole numbers? If so, name the triple family (e.g., 3-4-5).
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🧩 Example Problems (Based on Common Worksheet Layouts)
Let’s assume your worksheet has triangles like these:
#### 🔹 Problem 1:
Triangle with legs 6 and 8.
- Equation: $ 6^2 + 8^2 = c^2 $
- $ 36 + 64 = 100 $
- $ c = \sqrt{100} = 10 $
-
Answer: $ c = 10 $
-
Triple Family: Yes — 6-8-10 is a multiple of
3-4-5 (×2)
-
Triple: 3-4-5
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#### 🔹 Problem 2:
Triangle with leg 7 and hypotenuse 25.
- Let $ a = 7 $, $ c = 25 $, find $ b $
- $ 7^2 + b^2 = 25^2 $
- $ 49 + b^2 = 625 $
- $ b^2 = 576 $
- $ b = \sqrt{576} = 24 $
-
Answer: $ b = 24 $
-
Triple Family: Yes — 7-24-25 is a known triple
-
Triple: 7-24-25
---
#### 🔹 Problem 3:
Triangle with legs 9 and 12.
- $ 9^2 + 12^2 = c^2 $
- $ 81 + 144 = 225 $
- $ c = \sqrt{225} = 15 $
-
Answer: $ c = 15 $
-
Triple Family: Yes — 9-12-15 is a multiple of
3-4-5 (×3)
-
Triple: 3-4-5
---
#### 🔹 Problem 4:
Triangle with leg 15 and hypotenuse 25.
- $ a = 15 $, $ c = 25 $, find $ b $
- $ 15^2 + b^2 = 25^2 $
- $ 225 + b^2 = 625 $
- $ b^2 = 400 $
- $ b = \sqrt{400} = 20 $
-
Answer: $ b = 20 $
-
Triple Family: Yes — 15-20-25 is a multiple of
3-4-5 (×5)
-
Triple: 3-4-5
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💡 Tips:
- If the answer is not a whole number, simplify the radical.
- E.g., $ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} $
- Common triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-12-15 (multiple of 3-4-5), etc.
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📝 How to Fill Out Your Worksheet:
For each problem:
| # | Equation | Triple? | Triple Family |
|---|---------|--------|----------------|
| 1 | $ 6^2 + 8^2 = c^2 $ → $ c = 10 $ | Yes | 3-4-5 |
| 2 | $ 7^2 + b^2 = 25^2 $ → $ b = 24 $ | Yes | 7-24-25 |
| 3 | $ 9^2 + 12^2 = c^2 $ → $ c = 15 $ | Yes | 3-4-5 |
| 4 | $ 15^2 + b^2 = 25^2 $ → $ b = 20 $ | Yes | 3-4-5 |
> Replace these values with the actual side lengths from your image.
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✔ Final Advice:
Go through each triangle on your worksheet and follow the steps above. If you tell me the side lengths (e.g., "one leg is 12, hypotenuse is 13"), I can solve them one by one!
Would you like to list the side lengths from your worksheet, and I’ll help you solve each one step-by-step?
Parent Tip: Review the logic above to help your child master the concept of geometry worksheet and answers.
