Question

which set of side lengths forms a right triangle? 4, 6, 10 2, 3, 13 15, 36, 39 9, 12, 18

Explanation:

Step1: Check the first set of side lengths ($4,6,10$)

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. But \(4 + 6=10\), which does not satisfy the triangle inequality theorem. So, this cannot form a triangle, let alone a right - triangle.

Step2: Check the second set of side lengths ($2,3,13$)

Again, using the triangle inequality theorem, \(2+3 = 5<13\). So, this cannot form a triangle.

Step3: Check the third set of side lengths ($15,36,39$)

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the longest side). Calculate \(15^{2}+36^{2}=225 + 1296=1521\) and \(39^{2}=1521\). So, \(15^{2}+36^{2}=39^{2}\), which satisfies the Pythagorean theorem.

Step4: Check the fourth set of side lengths ($9,12,18$)

Calculate \(9^{2}+12^{2}=81 + 144 = 225\) and \(18^{2}=324\). Since \(9^{2}+12^{2}
eq18^{2}\), it does not satisfy the Pythagorean theorem.

Answer:

The set of side lengths \(15,36,39\) forms a right - triangle.