Question

1. if a = 7, b = 24, and c = 25, is the triangle a right triangle?

a. no, because a² + b² > c²
b. no, because a² + b² < c²
c. yes, because a² + b² = c²
d. cannot be determined

Explanation:

Step1: Calculate $a^{2}+b^{2}$

$a = 7$, so $a^{2}=7^{2}=49$; $b = 24$, so $b^{2}=24^{2}=576$. Then $a^{2}+b^{2}=49 + 576=625$.

Step2: Calculate $c^{2}$

$c = 25$, so $c^{2}=25^{2}=625$.

Step3: Compare

Since $a^{2}+b^{2}=625$ and $c^{2}=625$, we have $a^{2}+b^{2}=c^{2}$. According to the converse of the Pythagorean theorem, if $a^{2}+b^{2}=c^{2}$ in a triangle with side - lengths $a$, $b$, and $c$ (where $c$ is the longest side), then the triangle is a right - triangle.

Answer:

C. Yes, because $a^{2}+b^{2}=c^{2}$