Question

select all the correct answers. using the side lengths of the given triangles (not drawn to scale), determine which ones are right triangles

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side).

First triangle (yellow):

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

Second triangle (orange):

Let \(a = 6\), \(b = 22\). Then \(a^{2}+b^{2}=6^{2}+22^{2}=36+484 = 520\). If \(c = 23\), \(c^{2}=529\). Since \(6^{2}+22^{2}
eq23^{2}\), it is not a right - triangle.

Third triangle (light - blue):

Let \(a = 20\), \(b = 21\). Then \(a^{2}+b^{2}=20^{2}+21^{2}=400 + 441=841\). If \(c = 29\), \(c^{2}=841\). So \(20^{2}+21^{2}=29^{2}\), it is a right - triangle.

Fourth triangle (purple):

Let \(a = 8\), \(b = 15\). Then \(a^{2}+b^{2}=8^{2}+15^{2}=64+225 = 289\). If \(c = 17\), \(c^{2}=289\). So \(8^{2}+15^{2}=17^{2}\), it is a right - triangle.

Fifth triangle (blue):

Let \(a = 28\), \(b = 45\). Then \(a^{2}+b^{2}=28^{2}+45^{2}=784+2025 = 2809\). If \(c = 53\), \(c^{2}=2809\). So \(28^{2}+45^{2}=53^{2}\), it is a right - triangle.

Sixth triangle (green):

Let \(a = 5\), \(b = 11\). Then \(a^{2}+b^{2}=5^{2}+11^{2}=25 + 121=146\). If \(c = 14\), \(c^{2}=196\). Since \(5^{2}+11^{2}
eq14^{2}\), it is not a right - triangle.

Answer:

The right - triangles are the first (yellow: sides \(3\), \(4\), \(5\)), the third (light - blue: sides \(20\), \(21\), \(29\)), the fourth (purple: sides \(8\), \(15\), \(17\)), and the fifth (blue: sides \(28\), \(45\), \(53\)).