📐 Geometry · Similarity and Right Triangles

The Pythagorean Theorem practice questions

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Answer key with explanations

1. A right triangle has legs of length 9 and 12. How long is the hypotenuse?

   Answer: 15. The unknown side is the hypotenuse, so square the legs, add, and take the square root: 9² + 12² = 81 + 144 = 225, and c = √225 = 15. The choice 21 adds the legs instead of squaring them. The choice 225 is c²; the square root was forgotten. The choice 3√7 treats 12 as the hypotenuse and subtracts (12² − 9² = 63).

2. A right triangle has hypotenuse 19 and one leg 12. The other leg is b. Which equation is correct?

   Answer: b² + 12² = 19². The hypotenuse (19) must stand alone on its own side: b² + 12² = 19². Then b² = 361 − 144 = 217 and b = √217 ≈ 14.73. The choice 12² + 19² = b² treats the unknown leg as the hypotenuse. The choice b + 12 = 19 forgets to square. The choice b² + 19² = 12² puts the hypotenuse on the wrong side.

3. A right triangle has hypotenuse 13 and one leg 3. Find the other leg.

   Answer: 4√10 ≈ 12.65. A missing leg means big square minus small square: b² = 13² − 3² = 169 − 9 = 160. Since 160 = 16 × 10, b = √160 = 4√10 ≈ 12.65. The choice √178 adds the squares (169 + 9), which is right only when the hypotenuse is the unknown. The choice 10 subtracts the lengths (13 − 3). The choice 160 is b² with the square root forgotten.

4. A right triangle has legs 2 and 8. What is the hypotenuse, as a simplified radical and as a decimal?

   Answer: 2√17 ≈ 8.25. 2² + 8² = 4 + 64 = 68, so c = √68. Since 68 = 4 × 17, √68 = √4 × √17 = 2√17 ≈ 8.25. The choice 10 adds the legs. The choice 4√17 pulls out 4 instead of its root, 2. The choice 68 is c²; the square root was forgotten.

5. Which set of side lengths forms a right triangle?

   Answer: 12, 35, 37. Test each set with its longest side as c. For 12, 35, 37: 12² + 35² = 144 + 1225 = 1369, and 37² = 1369, so it is a right triangle. The others fail: 8² + 13² = 233 but 16² = 256; 13² + 14² = 365 but 19² = 361 (close, but not equal); 5² + 7² = 74 but 9² = 81.

6. A ramp rises 2 feet over a horizontal distance of 10 feet. How long is the sloped surface of the ramp, to the nearest hundredth of a foot?

   Answer: 2√26 ≈ 10.20 feet. The ground and the vertical rise meet at a right angle, so the sloped surface is the hypotenuse of a triangle with legs 2 and 10. 2² + 10² = 4 + 100 = 104, and 104 = 4 × 26, so the length is √104 = 2√26 ≈ 10.20 feet. The choice 12 adds the legs. The choice 8 subtracts them. The choice 104 is c²; the square root was forgotten.

7. A rectangular banner has a diagonal of 34 inches and a width of 16 inches. How tall is the banner?

   Answer: 30 inches. The diagonal is the hypotenuse and the width is one leg, so the height is the other leg: h² = 34² − 16² = 1156 − 256 = 900, and h = √900 = 30 inches. Check: 16² + 30² = 256 + 900 = 1156 = 34². The choice 18 subtracts the lengths (34 − 16). The choice 2√353 adds the squares (1156 + 256 = 1412). The choice 900 is h²; the square root was forgotten.

8. A right triangle has legs 3x and 4x and hypotenuse 35. What is x?

   Answer: 7. Put the hypotenuse alone: (3x)² + (4x)² = 35², so 9x² + 16x² = 1225 and 25x² = 1225. Then x² = 49, and since a length is positive, x = 7. The legs are 21 and 28, and 21² + 28² = 441 + 784 = 1225 = 35². The choice 5 adds instead of squaring (3x + 4x = 35). The choice 49 is x². The choice 5√7 forgets to square the coefficients (3x² + 4x² = 7x² = 1225).

9. A triangle has side lengths 3, 4 and √7. Which statement is true?

   Answer: It is a right triangle, and the hypotenuse is 4.. Since √7 ≈ 2.65, the longest side is 4, so 4 plays the role of c. Test: 3² + (√7)² = 9 + 7 = 16 = 4². So it is a right triangle with hypotenuse 4. The first choice calls the shortest side the hypotenuse. The second choice puts 4 on the wrong side of the test. The last choice is false: side lengths do not have to be whole numbers.

10. A rectangular park is 36 meters wide and 77 meters long. Dana walks from one corner to the opposite corner along the diagonal path. How much shorter is that path than walking along two sides of the park?

    Answer: 28 meters. Along two sides: 36 + 77 = 113 meters. The diagonal is the hypotenuse: 36² + 77² = 1296 + 5929 = 7225, and √7225 = 85 meters. The difference is 113 − 85 = 28 meters. The choice 85 is the length of the diagonal, not the difference. The choice 113 is the walk along two sides. The choice 41 subtracts the sides (77 − 36).