The Pythagorean theorem says that in a right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the two legs: a² + b² = c². In eighth grade, students explain a proof of the theorem and its converse (Common Core 8.G.B.6), use it to find missing side lengths in real-world and mathematical problems (8.G.B.7), and apply it to find the distance between two points on a coordinate plane (8.G.B.8).
These questions cover finding the hypotenuse, finding a missing leg, testing whether three lengths make a right triangle, ladders, screens and walking routes, and the distance between points. Start every problem by identifying the hypotenuse. If you are finding it, add the squares; if you are finding a leg, subtract the squares. Then take the square root. When an answer is not a whole number, round as the question asks. Each answer is checked instantly with a full worked solution, so students can see exactly which step went wrong.
Choose an option or type your answer, then press Check. Capital letters and extra spaces do not matter in typed answers, and units are optional.
Answer: A) 10 cm
c² = 6² + 8² = 36 + 64 = 100. c = square root of 100 = 10 cm.
Answer: 13
c² = 5² + 12² = 25 + 144 = 169. c = square root of 169 = 13.
Answer: B) 15
b² = 17² - 8² = 289 - 64 = 225. b = square root of 225 = 15.
Answer: 15
c² = 81 + 144 = 225. c = square root of 225 = 15.
Answer: B) 7, 24, 25
Test a² + b² = c² with the longest side as c. 7² + 24² = 49 + 576 = 625 = 25². The others fail, for example 4² + 5² = 41 but 6² = 36.
Answer: 12.2
c² = 49 + 100 = 149. The square root of 149 is about 12.207, which rounds to 12.2.
Answer: B) 8 feet
The ladder is the hypotenuse. Height² = 10² - 6² = 100 - 36 = 64, so the height is 8 feet.
Answer: 40 (also accepted: 40 inches)
The diagonal is the hypotenuse: d² = 32² + 24² = 1,024 + 576 = 1,600. d = 40 inches.
Answer: A) 5
The horizontal change is 4 - 1 = 3 and the vertical change is 6 - 2 = 4. Distance = square root of (9 + 16) = square root of 25 = 5.
Answer: 10
Horizontal change: 4 - (-2) = 6. Vertical change: 9 - 1 = 8. Distance = square root of (36 + 64) = square root of 100 = 10.
Answer: A) Yes, because 9² + 40² = 41²
9² + 40² = 81 + 1,600 = 1,681, and 41² = 1,681. The squares match, so by the converse of the theorem it is a right triangle.
Answer: 29 (also accepted: 29 cm)
The diagonal splits the rectangle into two right triangles. d² = 400 + 441 = 841, so d = 29 cm.
Answer: A) 7.5
b² = 81 - 25 = 56. The square root of 56 is about 7.48, which rounds to 7.5. (10.3 comes from adding the squares by mistake.)
Answer: 5 (also accepted: 5 miles)
North and east are at right angles, so the straight-line distance is the hypotenuse: square root of (9 + 16) = 5 miles.
When the hypotenuse is given and a leg is missing, subtract: b² = c² - a². Adding gives a leg longer than the hypotenuse, which is impossible.
a² + b² gives c², not c. For legs 6 and 8, c² = 100, so c = 10, not 100.
3 + 4 is not 5. The theorem works with squares: 9 + 16 = 25, and the square root of 25 is 5.
a² + b² = c² is only true when the triangle has a right angle. Check for the right angle first.
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Open the student portal →The hypotenuse is the side opposite the right angle. It is always the longest side of a right triangle.
If the squares of the two shorter sides add up to the square of the longest side, the triangle is a right triangle. It is how you test three lengths.