Identify the right triangles using the Pythagorean theorem.

Let’s solve each problem step by step using the Pythagorean theorem.

The Pythagorean theorem says:
In a right triangle, if the two shorter sides are called “a” and “b”, and the longest side (the hypotenuse) is “c”, then:
a² + b² = c²

We’ll check each triangle to see if this equation holds true. If it does, it’s a right triangle.

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Problem 1:

Sides: 6 ft, 13 ft, √205 ft

Check: Is 6² + 13² = (√205)²?

Calculate:
- 6² = 36
- 13² = 169
- Sum = 36 + 169 = 205
- (√205)² = 205

✔ Yes! So this is a right triangle.

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Problem 2:

Sides: 4 m, 7 m, √65 m

Check: Is 4² + 7² = (√65)²?

Calculate:
- 4² = 16
- 7² = 49
- Sum = 16 + 49 = 65
- (√65)² = 65

✔ Yes! So this is a right triangle.

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Problem 3:

Sides: 7 in, 8 in, 11 in

Which is the longest? → 11 in → so that should be “c”

Check: Is 7² + 8² = 11²?

Calculate:
- 7² = 49
- 8² = 64
- Sum = 49 + 64 = 113
- 11² = 121

✘ 113 ≠ 121 → Not equal → This is NOT a right triangle.

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Problem 4:

Sides: 5 ft, 12 ft, 13 ft

Longest side = 13 ft → that’s “c”

Check: Is 5² + 12² = 13²?

Calculate:
- 5² = 25
- 12² = 144
- Sum = 25 + 144 = 169
- 13² = 169

✔ Yes! So this is a right triangle.

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Problem 5:

Triangle XYZ with sides: XY = 12 ft, YZ = 16 ft, ZX = 20 ft

First, find the longest side → 20 ft → that’s the hypotenuse candidate.

Check: Is 12² + 16² = 20²?

Calculate:
- 12² = 144
- 16² = 256
- Sum = 144 + 256 = 400
- 20² = 400

✔ Yes! So triangle XYZ is a right triangle.

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Problem 6:

Triangle PQR with sides: PQ = 15 in, QR = 20 in, PR = 25 in

Longest side = 25 in → that’s “c”

Check: Is 15² + 20² = 25²?

Calculate:
- 15² = 225
- 20² = 400
- Sum = 225 + 400 = 625
- 25² = 625

✔ Yes! So triangle PQR is a right triangle.

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Final Answer:
1. Right triangle
2. Right triangle
3. Not a right triangle
4. Right triangle
5. Triangle XYZ is a right triangle
6. Triangle PQR is a right triangle