Question

30 which list of numbers could represent the side lengths of a right triangle? a 5, 10, 13 b 5, 12, 17 c 10, 24, 26

Explanation:

To determine if a set of numbers can represent the side lengths of a right triangle, we use the Pythagorean theorem, which states that for a right triangle with side lengths \(a\), \(b\) (the legs), and \(c\) (the hypotenuse, the longest side), \(a^2 + b^2 = c^2\).

Step 1: Check Option A (5, 10, 13)

Let \(a = 5\), \(b = 10\), and \(c = 13\) (since 13 is the longest side).
Calculate \(a^2 + b^2\): \(5^2 + 10^2 = 25 + 100 = 125\)
Calculate \(c^2\): \(13^2 = 169\)
Since \(125
eq 169\), this is not a right triangle.

Step 2: Check Option B (5, 12, 17)

Let \(a = 5\), \(b = 12\), and \(c = 17\) (17 is the longest side).
Calculate \(a^2 + b^2\): \(5^2 + 12^2 = 25 + 144 = 169\)
Calculate \(c^2\): \(17^2 = 289\)
Since \(169
eq 289\), this is not a right triangle.

Step 3: Check Option C (10, 24, 26)

Let \(a = 10\), \(b = 24\), and \(c = 26\) (26 is the longest side).
Calculate \(a^2 + b^2\): \(10^2 + 24^2 = 100 + 576 = 676\)
Calculate \(c^2\): \(26^2 = 676\)
Since \(10^2 + 24^2 = 26^2\), this satisfies the Pythagorean theorem.

Answer:

C. 10, 24, 26