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) 8, 14, 16
(b) 16, 30, 34
(c) 8, 11, 14
(d) 3, 4, 5

Explanation:

Step1: Recall Pythagorean theorem

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

Step2: Check (a)

Let \(a = 8\), \(b = 14\), \(c = 16\). Then \(a^{2}+b^{2}=8^{2}+14^{2}=64 + 196=260\) and \(c^{2}=16^{2}=256\). Since \(260
eq256\), it is not a right - triangle.

Step3: Check (b)

Let \(a = 16\), \(b = 30\), \(c = 34\). Then \(a^{2}+b^{2}=16^{2}+30^{2}=256+900 = 1156\) and \(c^{2}=34^{2}=1156\). Since \(a^{2}+b^{2}=c^{2}\), it is a right - triangle.

Step4: Check (c)

Let \(a = 8\), \(b = 11\), \(c = 14\). Then \(a^{2}+b^{2}=8^{2}+11^{2}=64 + 121=185\) and \(c^{2}=14^{2}=196\). Since \(185
eq196\), it is not a right - triangle.

Step5: Check (d)

Let \(a = 3\), \(b = 4\), \(c = 5\). Then \(a^{2}+b^{2}=3^{2}+4^{2}=9 + 16=25\) and \(c^{2}=5^{2}=25\). Since \(a^{2}+b^{2}=c^{2}\), it is a right - triangle.

Answer:

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