48 Pythagorean Theorem Worksheet with Answers [Word + PDF] - Free Printable

Let’s solve each problem step by step using the Pythagorean Theorem:
a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).

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Problem 1:

We have a right triangle with legs x and 16, and hypotenuse 20.

So:
x² + 16² = 20²
x² + 256 = 400
x² = 400 - 256 = 144
x = √144 = 12

✔ Final Answer for #1: 12

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Problem 2:

Legs are 6 and 2.5, find hypotenuse x.

x² = 6² + 2.5²
x² = 36 + 6.25 = 42.25
x = √42.25 = 6.5

✔ Final Answer for #2: 6.5

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Problem 3:

Legs are 12 and x, hypotenuse is 14.

12² + x² = 14²
144 + x² = 196
x² = 196 - 144 = 52
x = √52 ≈ 7.2 (rounded to nearest tenth)

Wait — let me check if they want exact or simplified? The problem says “find the length”, no rounding specified, but in #8 it says round to nearest tenth. Let’s keep it as √52 unless told otherwise… but looking at other problems, maybe simplify?

Actually, √52 = √(4×13) = 2√13 — but since others are decimals, probably decimal is fine. But let’s see what the context expects.

Looking at problem #8, it says “round to nearest tenth”. So maybe here too? But problem doesn’t specify. However, in school worksheets, often they expect simplified radical or decimal. Since 52 isn’t a perfect square, and others gave nice numbers, perhaps we leave as √52? Wait — let me recalculate:

14² = 196
12² = 144
196 - 144 = 52 → yes.

But maybe I made a mistake? Is 12 one leg, x the other, 14 hypotenuse? Yes.

Alternatively, maybe they want exact value? But in problem #6, it says “simplify your answer” — so maybe here too? Hmm.

Actually, let’s look back — problem #3 just says “x = ___”, no instruction. But in multiple choice #8, they ask to round. So perhaps for free response, exact is okay? But 52 is not nice.

Wait — maybe I misread the diagram? In problem 3, is 12 a leg and 14 the hypotenuse? Yes, from the drawing.

Perhaps they expect decimal? Let’s compute √52:

√49 = 7, √64=8, so √52 ≈ 7.211 → rounds to 7.2

I think for consistency with #8, we’ll go with 7.2

✔ Final Answer for #3: 7.2

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Problem 4: Check if sides 6, 8, 10 form a right triangle.

Check: 6² + 8² =? 10²
36 + 64 = 100 → 100 = 100 → YES

✔ Final Answer for #4: yes

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Problem 5: Sides 14, 16, 18

Check: 14² + 16² =? 18²
196 + 256 = 452
18² = 324 → 452 ≠ 324 → NO

Also, 14² + 18² = 196 + 324 = 520 vs 16²=256 → nope
16² + 18² = 256 + 324 = 580 vs 14²=196 → nope

So definitely not a right triangle.

✔ Final Answer for #5: no

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Problem 6: Ladder problem

Ladder forms hypotenuse. Distance from house = 6 ft (base), height on wall = 14 ft (height). Find ladder length (hypotenuse).

So:
x² = 6² + 14² = 36 + 196 = 232
x = √232

Simplify: √232 = √(4×58) = 2√58 — but problem says “simplify your answer”. Maybe they mean simplified radical? Or decimal?

In real-world context, usually decimal. But “simplify” might mean reduce radical.

√232 = √(4*58) = 2√58 — that’s simplified radical.

But sometimes “simplify” means approximate. Let’s see: √232 ≈ 15.23 → but problem didn’t say round.

Wait — in problem #8, it says “round to nearest tenth”, so here maybe not? But “simplify” could mean either.

Looking at typical textbook language: “simplify your answer” for radicals means write in simplest radical form. For decimals, they say “round”.

So I’ll go with 2√58

But let me double-check calculation:

6² = 36
14² = 196
Sum = 232 → correct.

√232 = √(4*58) = 2√58 — yes.

✔ Final Answer for #6: 2√58

*(If they wanted decimal, it would be about 15.2, but since it says “simplify”, radical form is better.)*

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Problem 7: Areas of squares on legs and hypotenuse.

Given: Area of square on hypotenuse c is 369.
Area of square on leg a is 81.
Find area of square on leg b.

By Pythagorean Theorem:
(area on a) + (area on b) = (area on c)
Because area of square on side = side², so a² + b² = c² → areas add up.

So:
81 + b_area = 369
b_area = 369 - 81 = 288

✔ Final Answer for #7: 288

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Problem 8: Multiple Choice — legs 3 ft and 4 ft, find hypotenuse x.

x² = 3² + 4² = 9 + 16 = 25
x = √25 = 5

Options:
a. 6.9 ft
b. 2.8 ft
c. 5.7 ft
d. 5 ft ← correct!

✔ Final Answer for #8: d. 5 ft

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Now compiling all final answers:

Final Answer:
1.) 12
2.) 6.5
3.) 7.2
4.) yes
5.) no
6.) 2√58
7.) 288
8.) d. 5 ft