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, 8, 9	○	○	○
(c) 27, 36, 45	○	○	○
(d) 6, 9, 11	○	○	○

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) 13, 20, 26

Calculate \(13^{2}+20^{2}=169 + 400=569\) and \(26^{2}=676\). Since \(13^{2}+20^{2}
eq26^{2}\), it is not a right - triangle.

Step3: Check (b) 4, 8, 9

Calculate \(4^{2}+8^{2}=16 + 64 = 80\) and \(9^{2}=81\). Since \(4^{2}+8^{2}
eq9^{2}\), it is not a right - triangle.

Step4: Check (c) 27, 36, 45

Calculate \(27^{2}+36^{2}=729+1296 = 2025\) and \(45^{2}=2025\). Since \(27^{2}+36^{2}=45^{2}\), it is a right - triangle.

Step5: Check (d) 6, 9, 11

Calculate \(6^{2}+9^{2}=36 + 81=117\) and \(11^{2}=121\). Since \(6^{2}+9^{2}
eq11^{2}\), it is not a right - triangle.

Answer:

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