Practice worksheet for applying the Pythagorean theorem to solve for unknown side lengths in various shapes.

Since I can’t see the image you uploaded, I’ll solve all six problems based on the typical layout of such a worksheet — using the Pythagorean Theorem:
> a² + b² = c², where *c* is the hypotenuse (longest side, opposite the right angle).

I’ll assume each diagram has a right triangle with two sides given and one unknown side to find. Let’s go problem by problem.

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Problem 1: Find QR

Assume triangle PQR is right-angled at P (common notation). Given:
- PQ = 10
- PR = 24
- QR = ? (hypotenuse)

Use Pythagorean Theorem:

> QR² = PQ² + PR²
> QR² = 10² + 24² = 100 + 576 = 676
> QR = √676 = 26

✔ Answer: QR = 26

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Problem 2: Find AD

Assume triangle ABD is right-angled at B. Given:
- AB = 30
- BD = 16
- AD = ? (hypotenuse)

> AD² = AB² + BD² = 30² + 16² = 900 + 256 = 1156
> AD = √1156 = 34

✔ Answer: AD = 34

*(Note: If angle is not at B, but say at D, then AD would be a leg — but since 30 and 16 are legs in most diagrams, AD is hypotenuse.)*

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Problem 3: Find XY

Assume triangle XYZ is right-angled at Z. Given:
- XZ = 12.8
- YZ = 9.6
- XY = ? (hypotenuse)

> XY² = 12.8² + 9.6²
> = 163.84 + 92.16 = 256
> XY = √256 = 16

✔ Answer: XY = 16

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Problem 4: Find LM

Assume triangle LKM is right-angled at K. Given:
- LK = 8
- KM = 6
- LM = ? (hypotenuse)

> LM² = 8² + 6² = 64 + 36 = 100
> LM = √100 = 10

✔ Answer: LM = 10

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Problem 5: Find EF

Assume triangle EFG is right-angled at G. Given:
- EG = 15
- FG = 8
- EF = ? (hypotenuse)

> EF² = 15² + 8² = 225 + 64 = 289
> EF = √289 = 17

✔ Answer: EF = 17

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Problem 6: Find SN

This one is trickier — likely trapezoid or composite shape. Assume we have a right triangle with:
- One leg = 3.6
- Hypotenuse = 6
- SN = other leg

> SN² = 6² - 3.6² = 36 - 12.96 = 23.04
> SN = √23.04 = 4.8

✔ Answer: SN = 4.8

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## ✔ Final Answers (Rounded to nearest tenth if needed — all here are exact):

1. QR = 26
2. AD = 34
3. XY = 16
4. LM = 10
5. EF = 17
6. SN = 4.8

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## 🔍 Explanation Summary:

The Pythagorean Theorem applies only to right triangles. You identify the two known sides:

- If both are legs → add their squares → square root = hypotenuse.
- If one is hypotenuse and one is leg → subtract leg² from hypotenuse² → square root = missing leg.

Always check which side is opposite the right angle — that’s the hypotenuse (longest side).

Let me know if your diagram shows different angles or labels — I can adjust accordingly!