Question

the pythagorean theorem states that
for all right triangles with legs a and b, and hypotenuse c
$a^2 + b^2 = c^2$
are these right triangles???
use the pythagorean theorem to check if the lengths could be the 3 sides of a right triangle or not.
remember, if the pythagorean theorem doesnt work, that means the triangle is not a right triangle.
show your work to justify your answer!!!

1. 12, 16, & 20
2. 7, 8, & 10
3. 24, 45, & 51

Explanation:

Step1: Analyze problem 16 (12, 16, 20)

Identify the hypotenuse (longest side, 20) and legs (12, 16). Apply Pythagorean theorem: \(a^2 + b^2 = c^2\). Calculate \(12^2 + 16^2 = 144 + 256 = 400\), and \(20^2 = 400\). Since \(12^2 + 16^2 = 20^2\), it's a right triangle.

Step2: Analyze problem 17 (7, 8, 10)

Hypotenuse is 10, legs 7 and 8. Calculate \(7^2 + 8^2 = 49 + 64 = 113\), and \(10^2 = 100\). Since \(7^2 + 8^2
eq 10^2\), it's not a right triangle.

Step3: Analyze problem 18 (24, 45, 51)

Hypotenuse is 51, legs 24 and 45. Calculate \(24^2 + 45^2 = 576 + 2025 = 2601\), and \(51^2 = 2601\). Since \(24^2 + 45^2 = 51^2\), it's a right triangle.

Answer:

1. Yes (12² + 16² = 20²)
2. No (7² + 8² ≠ 10²)
3. Yes (24² + 45² = 51²)