Pythagorean Theorem Word Problems Matching Worksheet Answer Key ... - Free Printable

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

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Problem 1:
Daniel rides 21 km west, then 18 km north. How far from start?
→ This forms a right triangle. Distance from start = hypotenuse.
So:
c² = 21² + 18² = 441 + 324 = 765
c = √765 ≈ 27.66 → rounds to 27.7 km
✔ Match with f

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Problem 2:
Square made of two triangles. Hypotenuse = 11 cm, width = 8 cm. Find height.
→ In a right triangle: a² + b² = c²
Here, width = 8, hypotenuse = 11, find height (other leg).
h² + 8² = 11² → h² + 64 = 121 → h² = 57 → h = √57 ≈ 7.55 cm
✔ Match with c

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Problem 3:
Equilateral triangle? Wait — equilateral triangles don’t have a “hypotenuse” unless it’s split into right triangles. But the problem says: “hypotenuse of an equilateral triangle that has a 13 cm leg and height of 7 cm.” That doesn’t make sense — equilateral triangles have all sides equal, so if leg is 13, all sides are 13. Height would be about 11.26, not 7. So likely typo — probably meant right triangle.

Assume it’s a right triangle with legs 13 cm and 7 cm? Or maybe base 13, height 7? Let’s read again: “hypotenuse of an equilateral triangle that has a 13 cm leg and a height of 7 cm.” Hmm. Maybe they mean a right triangle formed by splitting the equilateral? But height 7 and half-base? Not matching.

Wait — perhaps it’s just a right triangle with legs 13 and 7? Then hypotenuse = √(13² + 7²) = √(169+49)=√218≈14.76 — not in options.

Alternatively, maybe “leg” means one side, and height is perpendicular? If it’s a right triangle with base 13 and height 7, then hypotenuse = √(13² + 7²) = same as above.

But option i is 14.8 cm — close to 14.76. Maybe rounding.

Wait — let’s check Problem 9: similar wording. Maybe this is misstated. Let’s skip and come back.

Actually, looking at answer choices, 14.8 cm is option i. And √(13² + 7²) = √218 ≈ 14.76 → 14.8 cm. So probably they meant a right triangle with legs 13 and 7. Even though it says “equilateral”, likely error. We’ll go with that.

✔ Match with i

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Problem 4:
Roof is 51 ft long (hypotenuse?), base is 29 ft across. Find height.
→ Right triangle: roof = hypotenuse = 51, base = 29, find height.
h² + 29² = 51² → h² + 841 = 2601 → h² = 1760 → h = √1760 ≈ 41.95 → ~42 feet
✔ Match with e

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Problem 5:
Staircase for deck 12 ft high, starts 21 ft away. Length of staircase?
→ Right triangle: height=12, base=21, hypotenuse = length of stairs.
c² = 12² + 21² = 144 + 441 = 585 → c = √585 ≈ 24.186 → ~24.2 feet
✔ Match with g

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Problem 6:
Triangle paper: height 18 cm, width 10 cm. Find hypotenuse.
→ Assuming right triangle: legs 18 and 10.
c² = 18² + 10² = 324 + 100 = 424 → c = √424 ≈ 20.59 → ~20.6 cm
✔ Match with b

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Problem 7:
Hypotenuse of triangle with width 8 cm, height 12 cm.
→ Legs 8 and 12.
c² = 8² + 12² = 64 + 144 = 208 → c = √208 ≈ 14.422 → ~14.4 cm
✔ Match with h

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Problem 8:
Ladder 14 ft long, bottom 5 ft from building. How high up wall?
→ Right triangle: ladder = hypotenuse = 14, base = 5, find height.
h² + 5² = 14² → h² + 25 = 196 → h² = 171 → h = √171 ≈ 13.076 → ~13.1 feet
✔ Match with d

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Problem 9:
Width of triangle with height 15 cm, hypotenuse 19 cm.
→ Right triangle: height = 15, hypotenuse = 19, find width (other leg).
w² + 15² = 19² → w² + 225 = 361 → w² = 136 → w = √136 ≈ 11.66 — wait, not in options?

Options left: a. 11.7 cm — yes! √136 ≈ 11.66 → rounds to 11.7 cm
✔ Match with a

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Now let’s list matches:

1 → f
2 → c
3 → i
4 → e
5 → g
6 → b
7 → h
8 → d
9 → a

Leftover option: j? No, only a-i. All used.

Wait — we didn’t use option j? There is no j. Options are a through i.

We have 9 problems, 9 answers. Perfect.

Double-check Problem 3: we assumed right triangle with legs 13 and 7 → hypotenuse ≈14.8 → matches i. Even though it said “equilateral”, probably mistake. Given the numbers, it fits.

All others check out.

Final Answer:
1-f, 2-c, 3-i, 4-e, 5-g, 6-b, 7-h, 8-d, 9-a