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
(a) 5, 13, 14
(b) 22, 29, 36
(c) 9, 40, 41
(d) 18, 24, 30

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) 5, 13, 14

Calculate \(5^{2}+13^{2}=25 + 169=194\) and \(14^{2}=196\). Since \(194
eq196\), it's not a right - triangle.

Step3: Check (b) 22, 29, 36

Calculate \(22^{2}+29^{2}=484+841 = 1325\) and \(36^{2}=1296\). Since \(1325
eq1296\), it's not a right - triangle.

Step4: Check (c) 9, 40, 41

Calculate \(9^{2}+40^{2}=81 + 1600=1681\) and \(41^{2}=1681\). Since \(9^{2}+40^{2}=41^{2}\), it's a right - triangle.

Step5: Check (d) 18, 24, 30

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

Answer:

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