Pythagorean Theorem Worksheet For Each Triangle Find The Missing ... - Free Printable

Let’s solve each triangle 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:
Sides: 17 and ? , hypotenuse = 19
So:
?² + 17² = 19²
?² + 289 = 361
?² = 361 - 289 = 72
? = √72 ≈ 8.49 → But let’s check if it simplifies:
√72 = √(36×2) = 6√2 — but since this looks like a whole number problem, maybe we made a mistake? Wait — actually, 17-?-19 doesn’t form a nice triple. Let’s double-check calculation:

19² = 361
17² = 289
361 - 289 = 72 → correct.
But 72 isn’t a perfect square. Hmm… wait — perhaps I misread which side is which? In diagram 1, the legs are 17 and ?, hypotenuse 19 — yes. So answer is √72 or simplified 6√2. But maybe they want decimal? Or did I missee?

Wait — looking again at image description: Problem 1 has sides labeled 17, ?, 19 — with 19 as hypotenuse. So math is correct. But let’s hold on — maybe in original problem, numbers are different? No — user provided image description says:

“1. 19 / ? / 17” — so yes, 19 is hypotenuse.

Actually — let me recalculate:
19² = 361
17² = 289
Difference = 72 → sqrt(72) = 6√2 ≈ 8.485 — not integer. Maybe acceptable? But let’s check others.

Wait — perhaps I should assume all are integers? Let’s look at problem 2.

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Problem 2:
Legs: 5 and 12, find hypotenuse ?
5² + 12² = 25 + 144 = 169
√169 = 13 → nice! So ? = 13

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Problem 3:
Hypotenuse = 29, one leg = 20, find other leg ?
?² + 20² = 29²
?² + 400 = 841
?² = 841 - 400 = 441
√441 = 21 → perfect! So ? = 21

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Problem 4:
Legs: 9 and 24, find hypotenuse ?
9² + 24² = 81 + 576 = 657
Is 657 a perfect square?
25²=625, 26²=676 → no.
Wait — 9-24-? — divide by 3: 3-8-? → 3²+8²=9+64=73 → not helpful.
Wait — 9²=81, 24²=576, sum=657.
But 657 ÷ 9 = 73 → not square.
Wait — maybe I miscalculated? 24² is 576? Yes. 9²=81. Sum=657.
But 657 = 9 × 73 → not a perfect square. That can’t be — probably I misread the diagram.

Wait — user wrote for problem 4: “9 / 24 / ?” — and from context, 9 and 24 are legs, ? is hypotenuse. But 9-24-? should be multiple of 3-4-5? 3-4-5 scaled by 3 is 9-12-15 — not 24. Scaled by 8: 24-32-40 — not matching. Wait — 9 and 24 — GCD is 3 → 3 and 8 → 3²+8²=9+64=73 → so hypotenuse would be √(9*73)=3√73 — messy.

Wait — perhaps in the image, 24 is the hypotenuse? Let me re-read user input:

User said:
"4. 9 / 24 / ?" — and in typical labeling, if it's written as "9 / 24 / ?", and from diagram style, likely 9 and 24 are legs, ? is hypotenuse. But that gives non-integer.

Alternatively — maybe 24 is hypotenuse? Then:
9² + ?² = 24²
81 + ?² = 576
?² = 495 → still not nice.

Wait — another possibility: maybe it's 7-24-25 triple? But here it's 9 and 24.

Hold on — let me check standard triples:
Common ones: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-12-15, 12-16-20, etc.

9 and 24 — if we take 9-12-15, but 24 is not 12. Unless... wait, 9-24-? — let's compute numerically:
9² = 81
24² = 576
Sum = 657
√657 ≈ 25.63 — not integer.

But problem 1 also gave non-integer. That seems odd for homework. Perhaps I misassigned which side is which.

Let me reinterpret based on common diagram layouts:

In many textbooks, when they draw a right triangle and label three sides, the hypotenuse is usually the slanted one, and legs are horizontal and vertical.

For problem 1: labeled 19 (hypotenuse), 17 (base), ? (height) → so ? = √(19² - 17²) = √(361-289)=√72=6√2≈8.49

But maybe they expect exact form? Or perhaps there's a typo? Alternatively, maybe 19 is a leg? But that wouldn't make sense because 19>17, and if both legs, hypotenuse larger.

Wait — let's list all problems again with assumed roles:

Assume for each, the two given numbers are legs unless specified otherwise, but in diagrams, often the hypotenuse is labeled separately.

Looking back at user's text:

"1. 19 / ? / 17" — probably means sides are 19, ?, 17 with 19 being hypotenuse.

Similarly, "2. 5 / 12 / ?" — 5 and 12 legs, ? hypotenuse → 13

"3. 29 / ? / 20" — 29 hypotenuse, 20 leg, ? other leg → 21

"4. 9 / 24 / ?" — 9 and 24 legs, ? hypotenuse → √(81+576)=√657

But √657 can be simplified? 657 ÷ 9 = 73, so √657 = 3√73 — still not nice.

Wait — perhaps in problem 4, 24 is the hypotenuse? Then:

If 24 is hypotenuse, and 9 is one leg, then other leg ? = √(24² - 9²) = √(576 - 81) = √495 = √(9*55) = 3√55 — worse.

Another idea: maybe the numbers are switched. For example, in some diagrams, the vertical side is listed first. But without seeing image, hard to tell.

Perhaps for problem 4, it's 7-24-25, but written as 9 by mistake? Unlikely.

Wait — let's calculate √657 exactly: 25.632... not integer.

But let's check problem 1 again: √72 = 6√2 ≈8.485

Maybe the homework allows radicals or decimals? But typically for such problems, answers are integers.

Unless... I think I found the issue!

In problem 1: if the sides are 17 and ? , and hypotenuse 19, but 17 and 19 are close, difference small, so ? should be small — √72≈8.49, which is reasonable.

But let's verify with actual Pythagorean triples near there.

Perhaps the intended answer for 1 is 6√2, for 4 is 3√73, but that seems advanced.

Wait — another thought: in problem 4, maybe it's 9-40-41 or something, but 24 is given.

Let's do exact calculations for all:

Problem 1:
? = √(19² - 17²) = √(361 - 289) = √72 = √(36*2) = 6√2

Problem 2:
? = √(5² + 12²) = √(25+144) = √169 = 13

Problem 3:
? = √(29² - 20²) = √(841 - 400) = √441 = 21

Problem 4:
? = √(9² + 24²) = √(81 + 576) = √657

Now, simplify √657: factor 657.
657 ÷ 3 = 219
219 ÷ 3 = 73
73 is prime.
So 657 = 3² * 73
Thus √657 = 3√73

So answers are:
1. 6√2
2. 13
3. 21
4. 3√73

But perhaps the problem expects decimal approximations? Or maybe I have a fundamental mistake.

Wait — let's look back at the user's initial description: "You are an educational assistant..." and the image contains four triangles.

Perhaps in the image, for problem 4, the 24 is the hypotenuse? Let me try that.

If in problem 4, hypotenuse is 24, and one leg is 9, then other leg ? = √(24² - 9²) = √(576 - 81) = √495 = √(9*55) = 3√55 — still not nice.

If hypotenuse is ? , and legs 9 and 24, then ? = √657 as before.

Another possibility: maybe the numbers are 7 and 24 for problem 4? Because 7-24-25 is a famous triple. If it's 7 instead of 9, then ? = 25. But user said 9.

Perhaps it's a typo in my reading. Let me assume that for problem 4, it's meant to be 7-24-25, but since user wrote 9, I'll stick with calculation.

But let's check online or recall: is there a triple with 9 and 24? 9-12-15 is triple, but 24 is double 12, so if legs 9 and 24, not proportional.

9/3=3, 24/3=8, 3-8-√73, as before.

Perhaps the homework accepts simplified radical form.

So I'll go with:

1. 6√2
2. 13
3. 21
4. 3√73

But let's write them as requested.

Final Answer:
1. \( 6\sqrt{2} \)
2. 13
3. 21
4. \( 3\sqrt{73} \)

To confirm, let's box them.

Since the instruction is to provide final answer after reasoning, and no options are true, I'll just give the answers.

But perhaps for school level, they want numerical values? But 6√2 is approximately 8.49, 3√73≈25.63, but usually exact form is preferred.

I think for accuracy, I'll provide exact forms.

So:

Final Answer:
1. \( 6\sqrt{2} \)
2. 13
3. 21
4. \( 3\sqrt{73} \)

But let's make sure about problem 1: is 19 really the hypotenuse? In some diagrams, the largest number might not be hypotenuse, but in right triangle, hypotenuse is always longest side. 19>17, and ? will be less than 19, so yes.

Similarly for others.

So I think this is correct.