Grade 8 Math Practice Test: Pythagorean Theorem Word Problems and Distance
- Pythagorean Theorem Applications
- 10 questions
- 15 minutes
This test applies a2 + b2 = c2 to ladders, screens, maps, coordinate points and boxes. Tip: the hypotenuse is always the longest side and lies opposite the right angle; subtract squares when you know it and need a leg.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
A 13-foot ladder leans against a wall. Its base is 5 feet from the wall. How high up the wall does the ladder reach?
Correct answer: C
The ladder is the hypotenuse: h2 = 132 − 52 = 169 − 25 = 144, so h = 12 ft. 13.9 ft adds the squares as if the ladder were a leg.
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Question 2
A TV screen is 32 inches wide and 18 inches tall. What is the length of its diagonal, to the nearest tenth?
Correct answer: B
d2 = 322 + 182 = 1,024 + 324 = 1,348, and 1,348 ≈ 36.7 in. 26.5 in subtracts the squares, and 50 adds the sides.
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Question 3
What is the distance between the points (1, 2) and (7, 10)?
Correct answer: B
The horizontal change is 6 and the vertical change is 8, so d = 36 + 64 = 100 = 10. 14 adds the two changes.
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Question 4
What is the distance between the points (−3, 4) and (2, −8)?
Correct answer: C
The changes are 2 − (−3) = 5 and −8 − 4 = −12. d = 25 + 144 = 169 = 13. 17 adds 5 and 12.
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Question 5
Which set of side lengths forms a right triangle?
Correct answer: B
92 + 402 = 81 + 1,600 = 1,681 = 412, so by the converse of the theorem it is a right triangle. For 6, 8, 11: 36 + 64 = 100, not 121.
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Question 6
Ana walks 9 blocks east and then 12 blocks north. What is the straight-line distance from her starting point, in blocks?
Correct answer: D
d = 81 + 144 = 225 = 15 blocks. 21 is the distance she walked along the streets, not the straight-line distance.
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Question 7
A box is 3 in long, 4 in wide and 12 in tall. How long is the longest straight rod that fits inside, from one bottom corner to the opposite top corner?
Correct answer: A
The bottom diagonal is 9 + 16 = 5 in. With the height: 25 + 144 = 13 in. 5 in is only the diagonal of the bottom face.
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Question 8
A right triangle has a hypotenuse of 25 cm and one leg of 7 cm. How long is the other leg?
Correct answer: C
b2 = 625 − 49 = 576, so b = 24 cm. 26 cm comes from adding 49 to 625, which treats 25 as a leg.
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Question 9
The diagonal of a square is 10 cm. What is the length of a side, to the nearest tenth?
Correct answer: A
s2 + s2 = 100, so s2 = 50 and s = 50 ≈ 7.1 cm. 5 cm just halves the diagonal.
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Question 10
A rectangular field is 20 m long and 15 m wide. Leo walks from one corner to the opposite corner along the diagonal instead of along two edges. How many meters shorter is his path?
Correct answer: D
Along the edges: 20 + 15 = 35 m. The diagonal is 400 + 225 = 625 = 25 m. The difference is 35 − 25 = 10 m.
Questions you leave blank are not counted as wrong.
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