Question

1. which side lengths could form a right triangle? check all that apply. 1, 1, 2; 8, 15, 17; 18, 24, 30; 5, 9, 10; 12, 13, 14; 20, 25, 30

Explanation:

To determine if a set of side lengths can form 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), \(a^{2}+b^{2}=c^{2}\). We will check each set of side lengths:

Step 1: Check \(1, 1, 2\)

The longest side is \(2\). Calculate \(1^{2}+1^{2}=1 + 1=2\), and \(2^{2}=4\). Since \(2
eq4\), this does not form a right triangle.

Step 2: Check \(8, 15, 17\)

The longest side is \(17\). Calculate \(8^{2}+15^{2}=64 + 225 = 289\), and \(17^{2}=289\). Since \(8^{2}+15^{2}=17^{2}\), this forms a right triangle.

Step 3: Check \(18, 24, 30\)

The longest side is \(30\). Calculate \(18^{2}+24^{2}=324+576 = 900\), and \(30^{2}=900\). Since \(18^{2}+24^{2}=30^{2}\), this forms a right triangle.

Step 4: Check \(5, 9, 10\)

The longest side is \(10\). Calculate \(5^{2}+9^{2}=25 + 81=106\), and \(10^{2}=100\). Since \(106
eq100\), this does not form a right triangle.

Step 5: Check \(12, 13, 14\)

The longest side is \(14\). Calculate \(12^{2}+13^{2}=144 + 169=313\), and \(14^{2}=196\). Since \(313
eq196\), this does not form a right triangle.

Step 6: Check \(20, 25, 30\)

The longest side is \(30\). Calculate \(20^{2}+25^{2}=400+625 = 1025\), and \(30^{2}=900\). Since \(1025
eq900\), this does not form a right triangle.

Answer:

The side lengths that can form a right triangle are:

• \(8, 15, 17\)
• \(18, 24, 30\)