Question

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

(a) 10, 12, 16
(b) 9, 13, 16
(c) 15, 36, 39
(d) 5, 12, 13

Explanation:

Step1: Recall the Pythagorean theorem

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

Step2: Check part (a)

Let \(a = 10\), \(b = 12\), \(c=16\).
\(a^{2}+b^{2}=10^{2}+12^{2}=100 + 144=244\)
\(c^{2}=16^{2}=256\)
Since \(244
eq256\), it is not a right - triangle.

Step3: Check part (b)

Let \(a = 9\), \(b = 13\), \(c = 16\)
\(a^{2}+b^{2}=9^{2}+13^{2}=81+169 = 250\)
\(c^{2}=16^{2}=256\)
Since \(250
eq256\), it is not a right - triangle.

Step4: Check part (c)

Let \(a = 15\), \(b = 36\), \(c = 39\)
\(a^{2}+b^{2}=15^{2}+36^{2}=225+1296=1521\)
\(c^{2}=39^{2}=1521\)
Since \(1521 = 1521\), it is a right - triangle.

Step5: Check part (d)

Let \(a = 5\), \(b = 12\), \(c = 13\)
\(a^{2}+b^{2}=5^{2}+12^{2}=25 + 144=169\)
\(c^{2}=13^{2}=169\)
Since \(169=169\), it is a right - triangle.

Answer:

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