Pythagorean Theorem practice worksheet with 12 problems involving right triangles and side lengths.

Sure! Let’s solve each problem step by step. The worksheet is about the Pythagorean Theorem, which states:

> In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
> a² + b² = c², where c is the hypotenuse.

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## Part 1: Do the following lengths form a right triangle?

We check if a² + b² = c², where c is the longest side.

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1) Sides: 6, 8, 8

Longest side = 8 → Check: 6² + 8² =? 8²
→ 36 + 64 = 100 ≠ 64 → ✘ Not a right triangle

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2) Sides: 5, 12, 13

Longest side = 13 → Check: 5² + 12² =? 13²
→ 25 + 144 = 169 = 13² → ✔ Yes, it is a right triangle

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3) Sides: 6, 8, 10

Longest side = 10 → Check: 6² + 8² =? 10²
→ 36 + 64 = 100 = 10² → ✔ Yes, it is a right triangle

*(This is a classic 3-4-5 scaled up by 2: 6-8-10)*

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4) Sides: 3, 4, 5

Longest side = 5 → Check: 3² + 4² =? 5²
→ 9 + 16 = 25 = 5² → ✔ Yes, it is a right triangle

*(The most famous Pythagorean triple!)*

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5) a = 6.4, b = 12, c = 12.2

Longest side = 12.2 → Check: 6.4² + 12² =? 12.2²
→ 40.96 + 144 = 184.96
→ 12.2² = 148.84 → Wait, that can’t be right.

Let me recalculate:

6.4² = 40.96
12² = 144
Sum = 40.96 + 144 = 184.96

12.2² = 12.2 × 12.2 = (12 + 0.2)² = 144 + 4.8 + 0.04 = 148.84? → That’s wrong.

Actually:

12.2 × 12.2:
= (12 + 0.2)(12 + 0.2)
= 144 + 2×12×0.2 + 0.04
= 144 + 4.8 + 0.04 = 148.84

Wait — 184.96 ≠ 148.84 → So not equal.

But hold on — we assigned c = 12.2 as the hypotenuse, but 12 > 12.2? No, 12.2 is larger than 12. So c=12.2 is correct.

But 6.4² + 12² = 40.96 + 144 = 184.96
12.2² = 148.84 → That’s less than 184.96 → so not equal.

✔ Actually, I made a calculation error earlier.

Let me compute 12.2² properly:

12.2 × 12.2:

Breakdown:
12 × 12 = 144
12 × 0.2 = 2.4 → doubled = 4.8 (since two cross terms)
0.2 × 0.2 = 0.04
Total = 144 + 4.8 + 0.04 = 148.84

But 6.4² + 12² = 40.96 + 144 = 184.96

So 184.96 ≠ 148.84 → ✘ Not a right triangle

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6) a = 2.1, b = 7.2, c = 7.5

Longest side = 7.5 → Check: 2.1² + 7.2² =? 7.5²

Calculate:

2.1² = 4.41
7.2² = 51.84
Sum = 4.41 + 51.84 = 56.25

7.5² = 56.25 → ✔ Yes, it is a right triangle

*(This is a scaled version of 7-24-25? Let’s see: 2.1 : 7.2 : 7.5 → divide by 0.3 → 7 : 24 : 25 → yes!)*

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## Part 2: Find each missing length to the nearest tenth

We use a² + b² = c², solving for the unknown.

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7) Right triangle: legs 4 and ?, hypotenuse 8

Let missing leg = x
→ 4² + x² = 8²
→ 16 + x² = 64
→ x² = 48
→ x = √48 ≈ 6.928 → ≈ 6.9

✔ Answer: 6.9

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8) Right triangle: legs 3 and ?, hypotenuse 5

Let missing leg = x
→ 3² + x² = 5²
→ 9 + x² = 25
→ x² = 16
→ x = 4.0

✔ Answer: 4.0

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9) Right triangle: leg 7, hypotenuse 10, find other leg

Let missing leg = x
→ 7² + x² = 10²
→ 49 + x² = 100
→ x² = 51
→ x = √51 ≈ 7.141 → ≈ 7.1

✔ Answer: 7.1

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10) Right triangle: leg 3, leg 7, find hypotenuse

Let hypotenuse = c
→ 3² + 7² = c²
→ 9 + 49 = 58
→ c = √58 ≈ 7.615 → ≈ 7.6

✔ Answer: 7.6

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11) Right triangle: leg 2, hypotenuse 7, find other leg

Let missing leg = x
→ x² + 2² = 7²
→ x² + 4 = 49
→ x² = 45
→ x = √45 ≈ 6.708 → ≈ 6.7

✔ Answer: 6.7

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12) Right triangle: leg 2, hypotenuse 6, find other leg

Let missing leg = x
→ x² + 2² = 6²
→ x² + 4 = 36
→ x² = 32
→ x = √32 ≈ 5.656 → ≈ 5.7

✔ Answer: 5.7

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## ✔ Final Answers:

Part 1: Right Triangle?

1) ✘ No
2) ✔ Yes
3) ✔ Yes
4) ✔ Yes
5) ✘ No
6) ✔ Yes

Part 2: Missing Lengths (to nearest tenth)

7) 6.9
8) 4.0
9) 7.1
10) 7.6
11) 6.7
12) 5.7

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Let me know if you’d like a diagram or further explanation for any problem!