Practice problems for the Pythagorean Theorem, solving for missing sides in right triangles.

Let’s solve each problem using the Pythagorean Theorem:
In a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle), and a and b are the other two sides.

We’ll go one by one.

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1. Triangle ABC, right-angled at B.
AB = 12, BC = 5, find AC (hypotenuse).
AC² = AB² + BC² = 12² + 5² = 144 + 25 = 169
AC = √169 = 13

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2. Right-angled at B. AB = 4, BC = 3, find AC.
AC² = 4² + 3² = 16 + 9 = 25
AC = √25 = 5

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3. Right-angled at B. AB = 6, BC = 8, find AC.
AC² = 6² + 8² = 36 + 64 = 100
AC = √100 = 10

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4. Right-angled at B. AB = 24, BC = 7, find AC (hypotenuse).
AC² = 24² + 7² = 576 + 49 = 625
AC = √625 = 25

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5. Right-angled at B. AB = 12, BC = 9, find AC (hypotenuse).
AC² = 12² + 9² = 144 + 81 = 225
AC = √225 = 15

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6. Right-angled at B. AB = 15, BC = 8, find AC (hypotenuse).
AC² = 15² + 8² = 225 + 64 = 289
AC = √289 = 17

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7. Right-angled at B. AB = 12, AC = 20 (hypotenuse), find BC.
BC² = AC² - AB² = 400 - 144 = 256
BC = √256 = 16

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8. Right-angled at B. BC = 5, AC = 13 (hypotenuse), find AB.
AB² = AC² - BC² = 169 - 25 = 144
AB = √144 = 12

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9. Right-angled at B. BC = 18, AC = 30 (hypotenuse), find AB.
AB² = 30² - 18² = 900 - 324 = 576
AB = √576 = 24

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10. Right-angled at B. AB = 24, AC = 40 (hypotenuse), find BC.
BC² = 40² - 24² = 1600 - 576 = 1024
BC = √1024 = 32

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11. Right-angled at B. BC = 80, AC = 100 (hypotenuse), find AB.
AB² = 100² - 80² = 10000 - 6400 = 3600
AB = √3600 = 60

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12. Right-angled at B. AB = 68, AC = 85 (hypotenuse), find BC.
BC² = 85² - 68² = 7225 - 4624 = 2601
BC = √2601 = 51

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13. Right-angled at B. AB = 9, BC = 7, find AC (hypotenuse).
AC² = 9² + 7² = 81 + 49 = 130
AC = √130 ≈ 11.40 (but since others are whole numbers, maybe check — wait, 130 isn’t perfect square. But problem says “AC = ?”, so we leave as √130 or decimal? Let’s keep exact unless told otherwise. But looking at pattern, maybe they expect simplified radical? Actually, let’s compute numerically if needed. But in school context, often leave as √130. However, let me double-check: 9²=81, 7²=49, sum=130 → yes. So AC = √130. But perhaps I made mistake? No. Maybe it's acceptable. Wait — actually, let’s see problem 14 and 15 — they have non-right angles marked? No, all have right angle symbol. So for 13, answer is √130. But let’s write as decimal rounded? Problem doesn’t specify. Since others are integers, maybe I misread? Let me check image description again — no, user didn’t provide image, but based on standard problems, perhaps it’s meant to be integer? Wait, 9 and 7 — 81+49=130 — not perfect square. So we’ll report as √130 or approx 11.4. But to match format, perhaps write exact value. However, looking at problem 14 and 15, they also may not be perfect squares. Let’s proceed.

Actually, let’s calculate all and see. For now, AC = √130. But maybe the problem expects us to leave it like that. Alternatively, perhaps I should compute decimal. But in homework, often exact form. Let’s do exact unless specified.

Wait — problem 14: AB=?, with sides 12 and 19? Wait, let’s read carefully.

Problem 14: Right-angled at B. AB = ?, BC = 12, AC = 19? Wait, no: diagram shows A to C is 19, B to C is 12, right angle at B, so AB and BC are legs, AC hypotenuse? But 19 is labeled on AC? Yes. So AB² + BC² = AC² → AB² + 12² = 19² → AB² = 361 - 144 = 217 → AB = √217. Similarly, problem 15: AC=?, AB=15, BC=11, right angle at B, so AC² = 15² + 11² = 225 + 121 = 346 → AC = √346.

But these are not integers. However, in many such worksheets, they might expect decimal approximations or exact radicals. Since the instruction is to solve accurately, and no specification, I’ll provide exact values, but for consistency with earlier problems which were integers, perhaps I made an error in interpretation.

Wait — let me re-express:

For problem 13: vertices A, B, C. Right angle at B. So sides: AB and BC are legs, AC hypotenuse. Given AB=9, BC=7, so AC=√(81+49)=√130.

Similarly, problem 14: right angle at B. Sides: AB=?, BC=12, AC=19. So AB = √(19² - 12²) = √(361-144)=√217.

Problem 15: right angle at B. AB=15, BC=11, AC=?. So AC=√(225+121)=√346.

Problem 16: right angle at B. AB=?, BC=5, AC=12. So AB=√(144-25)=√119.

Problem 17: right angle at B. AB=?, BC=19, AC=23. So AB=√(529-361)=√168=√(4*42)=2√42.

Problem 18: right angle at B. AB=?, BC=21, AC=34. So AB=√(1156-441)=√715.

These are messy. But perhaps in the original image, some are different? Since I don't have image, I must go by text description.

Alternatively, maybe for problems 13-18, the right angle is not always at B? Let me check the labels.

In problem 13: "A \ AC = ? / 9 \ B \ 7 / C" and right angle at B — so yes, legs AB=9, BC=7, hypotenuse AC.

Similarly, 14: "A \ AB = ? / 19 \ B \ 12 / C" — right angle at B, so AB and BC are legs, AC=19 hypotenuse. So AB = √(19² - 12²) = √217.

But perhaps the worksheet expects decimal answers? Or maybe I should leave as simplified radicals.

To be safe, I'll compute numerical values rounded to two decimals where necessary, but for problems 1-12, they are integers, so for 13-18, I'll provide exact radicals or decimals as appropriate.

But let's see the pattern: problems 1-12 are all Pythagorean triples or simple, 13-18 are not, so likely they want exact form or decimal.

Since the user said "solve accurately", and no specification, I'll provide exact values for all, but for final answer, list them.

But for consistency, perhaps the expected answer for 13 is √130, etc.

However, in many school contexts, they might expect the student to calculate the decimal. Let me calculate:

13: √130 ≈ 11.40

14: √217 ≈ 14.73

15: √346 ≈ 18.60

16: √119 ≈ 10.91

17: √168 = √(4*42) = 2√42 ≈ 12.96

18: √715 ≈ 26.74

But this seems messy. Perhaps I misidentified the hypotenuse in some cases.

Let me double-check problem 14: "A \ AB = ? / 19 \ B \ 12 / C" — and right angle at B. So from A to C is 19, which is the hypotenuse. Yes.

Another possibility: in some diagrams, the right angle might be at a different vertex, but the text says "right angle at B" for all? In the initial description, for 1-12, it's clear, for 13-18, the user didn't specify, but from the way it's written, likely right angle at B for all.

Perhaps for problem 16: "B \ 5 / A \ 12 / C" and AB = ?, with right angle at B. So sides: AB and BC are legs, AC=12 hypotenuse? But 12 is labeled on AC? In text: "A \ 12 / C" and "B \ 5 /" and "AB = ?", so likely AC=12, BC=5, find AB. So AB = √(12² - 5²) = √(144-25)=√119, same as before.

I think it's correct. So I'll proceed with exact values, but for final answer, since the first 12 are integers, and the last 6 are not, perhaps the worksheet has typos or I need to output as is.

Maybe for 13-18, they want the length, so I'll provide the exact radical form.

But to match the format, let's list all answers.

Perhaps the user expects only the numerical value, so for non-integers, use decimal.

But in the interest of time, and since this is a common type, I'll provide the answers as calculated.

Let me summarize:

1. 13
2. 5
3. 10
4. 25
5. 15
6. 17
7. 16
8. 12
9. 24
10. 32
11. 60
12. 51
13. √130 or approximately 11.40
14. √217 or approximately 14.73
15. √346 or approximately 18.60
16. √119 or approximately 10.91
17. √168 = 2√42 or approximately 12.96
18. √715 or approximately 26.74

But for the final answer, since the instruction is to provide the final correct answer, and it's multiple problems, I need to list all.

Perhaps the user wants each answer separately, but the response format says "Final Answer" section, so likely list them.

Since it's a worksheet, probably expect the values.

To be precise, I'll provide the exact values for 13-18, but in many educational contexts, they simplify or leave as radical.

For 17, √168 can be simplified to 2√42, but perhaps not necessary.

I think for accuracy, I'll use the exact radical form for 13-18, but since the first 12 are integers, and the problem might expect numerical, let's see the context.

Another thought: in problem 13, if AB=9, BC=7, and right angle at B, then AC=√(81+49)=√130, which is correct.

Similarly for others.

So I'll go with that.

For the final answer, I'll list all 18 answers.

But the response format says "Final Answer" with the answer, so perhaps box them or list.

Since it's multiple, I'll write them in order.

Let me write the final answers clearly.

Final Answer:
1. 13
2. 5
3. 10
4. 25
5. 15
6. 17
7. 16
8. 12
9. 24
10. 32
11. 60
12. 51
13. √130
14. √217
15. √346
16. √119
17. 2√42
18. √715

But for 17, √168 = √(4*42) = 2√42, which is simplified.

For others, radicals are already simplified.

To make it neat, perhaps use decimals for 13-18, but I think exact is better.

Since the user might expect numerical values, and to match the integer style, but it's not possible, so I'll keep exact.

I think this is accurate.

So the final answer is the list above.