Question

which of the following lengths could be used to make a right triangle?
6, 7, 8
9, 10, 12
12, 13, 14
6, 8, 10

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, the sum of the squares of the two shorter sides (legs) should equal the square of the longest side (hypotenuse), i.e., \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

Step2: Check option 6,7,8

Calculate \(6^2 + 7^2 = 36 + 49 = 85\), and \(8^2 = 64\). Since \(85
eq64\), not a right triangle.

Step3: Check option 9,10,12

Calculate \(9^2 + 10^2 = 81 + 100 = 181\), and \(12^2 = 144\). Since \(181
eq144\), not a right triangle.

Step4: Check option 12,13,14

Calculate \(12^2 + 13^2 = 144 + 169 = 313\), and \(14^2 = 196\). Since \(313
eq196\), not a right triangle.

Step5: Check option 6,8,10

Calculate \(6^2 + 8^2 = 36 + 64 = 100\), and \(10^2 = 100\). Since \(100 = 100\), this satisfies the Pythagorean theorem.

Answer:

6, 8, 10 (the last option: 6, 8, 10)