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) 6, 14, 19	○	○	○
(c) 3, 9, 10	○	○	○
(d) 5, 10, 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) 20, 21, 29

Calculate \(20^{2}+21^{2}=400 + 441=841\) and \(29^{2}=841\). Since \(20^{2}+21^{2}=29^{2}\), it is a right - triangle.

Step3: Check (b) 6, 14, 19

Calculate \(6^{2}+14^{2}=36+196 = 232\) and \(19^{2}=361\). Since \(6^{2}+14^{2}
eq19^{2}\), it is not a right - triangle.

Step4: Check (c) 3, 9, 10

Calculate \(3^{2}+9^{2}=9 + 81=90\) and \(10^{2}=100\). Since \(3^{2}+9^{2}
eq10^{2}\), it is not a right - triangle.

Step5: Check (d) 5, 10, 11

Calculate \(5^{2}+10^{2}=25 + 100=125\) and \(11^{2}=121\). Since \(5^{2}+10^{2}
eq11^{2}\), it is not a right - triangle.

Answer:

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