Question

determine whether a triangle with the given side lengths is a right triangle.

side lengths	right triangle	not a right triangle	not enough information
(b) 4, 7, 8	○	○	○
(c) 18, 32, 40	○	○	○
(d) 7, 24, 25	○	○	○

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), \(a^{2}+b^{2}=c^{2}\).

Step2: Check (a) 18, 24, 30

Calculate \(18^{2}+24^{2}=324 + 576=900\) and \(30^{2}=900\). Since \(18^{2}+24^{2}=30^{2}\), it is a right - triangle.

Step3: Check (b) 4, 7, 8

Calculate \(4^{2}+7^{2}=16 + 49 = 65\) and \(8^{2}=64\). Since \(4^{2}+7^{2}
eq8^{2}\), it is not a right - triangle.

Step4: Check (c) 18, 32, 40

Calculate \(18^{2}+32^{2}=324+1024 = 1348\) and \(40^{2}=1600\). Since \(18^{2}+32^{2}
eq40^{2}\), it is not a right - triangle.

Step5: Check (d) 7, 24, 25

Calculate \(7^{2}+24^{2}=49 + 576=625\) and \(25^{2}=625\). Since \(7^{2}+24^{2}=25^{2}\), it is a right - triangle.

Answer:

(a) Right triangle
(b) Not a right triangle
(c) Not a right triangle
(d) Right triangle