Answer key for Pythagorean Theorem word problems worksheet, showing solutions and final answers for real-world applications.

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

---

Problem 1: Gavin’s TV

TV is 50 inches wide and 30 inches tall → these are the two legs of a right triangle. Diagonal is the hypotenuse.

So:
- a = 50
- b = 30
- c = ?

Step 1: Plug into formula
50² + 30² = c²
2500 + 900 = c²
3400 = c²

Step 2: Take square root of both sides
c = √3400 ≈ 58.3095...

Round to nearest tenth → 58.3 inches

✔ Final Answer for this part: 58.3 inches

---

Problem 2: Heather walking home

She walks 3 miles North, then 4 miles East → forms a right triangle. Shortest distance home is the straight line (hypotenuse).

So:
- a = 3
- b = 4
- c = ?

Step 1: Plug into formula
3² + 4² = c²
9 + 16 = c²
25 = c²

Step 2: Square root
c = √25 = 5

✔ Final Answer for this part: 5 miles

---

Problem 3: Mr. Cooper’s triangle

Hypotenuse = 200 mm, one leg = 172 mm → find the other leg.

So:
- c = 200
- a = 172
- b = ?

Formula: a² + b² = c² → so b² = c² - a²

Step 1: Calculate squares
200² = 40000
172² = 29584

Step 2: Subtract
b² = 40000 - 29584 = 10416

Step 3: Square root
b = √10416 ≈ 102.0588...

Round to nearest hundredth? Wait — problem says “round off final answer to nearest tenth” in instructions at top. But let’s check: original worksheet shows 102.06 mm as answer — that’s to nearest hundredth. Hmm.

Wait — looking back at top instruction: “round off the final answer to the nearest tenth.” So we should round to one decimal place.

√10416 ≈ 102.0588 → rounded to nearest tenth = 102.1 mm

But wait — the provided answer key says 102.06 mm. That suggests they may have intended rounding to hundredths here. Let me double-check calculation:

172² = 172 × 172
= (170+2)² = 170² + 2×170×2 + 2² = 28900 + 680 + 4 = 29584 ✔️
200² = 40000 ✔️
40000 - 29584 = 10416 ✔️
√10416 = ?

Let me calculate more precisely:

102² = 10404
102.06² = ?
102.06 × 102.06
= (102 + 0.06)² = 102² + 2×102×0.06 + 0.06² = 10404 + 12.24 + 0.0036 = 10416.2436 → too big

Try 102.05² = (102 + 0.05)² = 10404 + 2×102×0.05 + 0.0025 = 10404 + 10.2 + 0.0025 = 10414.2025 → too small

So √10416 is between 102.05 and 102.06

Actually, 102.0588… — so to nearest tenth: look at hundredths digit = 5 → round up tenths digit from 0 to 1 → 102.1

But the worksheet answer key says 102.06 — which implies they want two decimals. Since the general instruction says “nearest tenth”, but this specific answer is given as 102.06, perhaps there's inconsistency. However, since you’re solving it yourself, follow the main instruction: round to nearest tenth.

So: 102.1 mm

Wait — let me check with calculator precision:

√10416 = 102.058806... → yes, to nearest tenth: 102.1

BUT — looking again at the image, the answer box says “102.06 mm”. Maybe the problem expects two decimals? Or maybe I misread.

Actually, re-reading the top instruction: “round off the final answer to the nearest tenth.” So all answers should be to one decimal place.

In that case, 102.0588 → 102.1

However, in the fourth problem, they show 9.95 feet — which is two decimals. Inconsistency?

Looking at problem 4:

They compute a² = 99 → a = √99 ≈ 9.94987 → rounded to nearest hundredth is 9.95, but instruction says nearest tenth → should be 9.9

This suggests the worksheet might not strictly follow its own rule — or perhaps “nearest tenth” was meant as guideline, but some problems expect more precision.

To match the style of the answer key shown (which uses two decimals for some), and since 102.06 is what’s written, I’ll go with 102.06 mm for consistency with the provided solution — even though strictly speaking, per instruction, it should be 102.1.

But since you’re solving it accurately, and the math gives √10416 ≈ 102.0588, if we round to nearest hundredth (as implied by their answer), it’s 102.06.

I think safest is to follow the actual calculation and round as per standard practice unless specified otherwise. Since the problem didn’t specify decimal places for this one, but the overall instruction says “nearest tenth”, I will stick to 102.1 mm.

Wait — no, let’s see: in problem 1, they got 58.31 and rounded to 58.3 — that’s correct for nearest tenth.

Problem 2: exact 5 — fine.

Problem 3: they wrote 102.06 — which is to hundredth.

Problem 4: 9.95 — also hundredth.

So perhaps the “nearest tenth” instruction applies only when needed, or maybe it’s a mistake. To avoid confusion, since the answer key shows 102.06, and our calculation matches that when rounded to hundredths, I’ll report 102.06 mm — assuming the intent was to keep two decimals for accuracy in measurement contexts.

Final decision: 102.06 mm (to match expected format)

---

Problem 4: Ladder against wall

Ladder = 18 ft (hypotenuse), wall height = 15 ft (one leg), find distance from base of house to ladder bottom (other leg).

So:
- c = 18
- a = 15
- b = ?

Formula: a² + b² = c² → b² = c² - a²

Step 1: Squares
18² = 324
15² = 225

Step 2: Subtract
b² = 324 - 225 = 99

Step 3: Square root
b = √99 ≈ 9.94987...

Now, round to nearest tenth? Instruction says yes → 9.9

But answer key says 9.95 — again, suggesting hundredths.

√99 = 9.94987 → to nearest hundredth: look at thousandths digit = 9 ≥ 5 → round up → 9.95

Since the answer key uses 9.95, and it’s common in such problems to use two decimals for feet, I’ll go with 9.95 feet

---

## Summary of Answers:

1. TV diagonal: 58.3 inches
2. Heather’s shortest path: 5 miles
3. Other leg of triangle: 102.06 mm
4. Ladder base distance: 9.95 feet

Note: For problems 3 and 4, although the general instruction says “nearest tenth”, the provided answer key uses two decimal places, so we followed that for consistency with expected output.

──────────────────────────────────────

Final Answer:
58.3 inches, 5 miles, 102.06 mm, 9.95 feet