Question

which set of numbers are possible side lengths for a right triangle? a 1,2,3 b 3,5,7 c 6,8,10 d 10,12,20

Explanation:

To determine which set of numbers can be 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 (1, 2, 3)

Calculate \(1^2 + 2^2\) and compare it to \(3^2\).
\(1^2 + 2^2 = 1 + 4 = 5\)
\(3^2 = 9\)
Since \(5
eq 9\), this is not a right triangle.

Step 2: Check Option B (3, 5, 7)

Calculate \(3^2 + 5^2\) and compare it to \(7^2\).
\(3^2 + 5^2 = 9 + 25 = 34\)
\(7^2 = 49\)
Since \(34
eq 49\), this is not a right triangle.

Step 3: Check Option C (6, 8, 10)

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

Step 4: Check Option D (10, 12, 20)

Calculate \(10^2 + 12^2\) and compare it to \(20^2\).
\(10^2 + 12^2 = 100 + 144 = 244\)
\(20^2 = 400\)
Since \(244
eq 400\), this is not a right triangle.

Answer:

C. 6, 8, 10