Question

start up: john drew a triangle with side lengths of 5, 12, and 13. his friend, bryan, looked at it and asked john if it was a right triangle. john’s response was yes. (pythagorean theorem: $a^{2}+b^{2}=c^{2}$) explain or show how john can prove to bryan that the triangle is a right triangle.

Explanation:

Step1: Identify sides for Pythagorean Theorem

Let $a = 5$, $b = 12$, $c = 13$.

Step2: Calculate $a^{2}+b^{2}$

$a^{2}+b^{2}=5^{2}+12^{2}=25 + 144=169$.

Step3: Calculate $c^{2}$

$c^{2}=13^{2}=169$.

Step4: Compare results

Since $a^{2}+b^{2}=169$ and $c^{2}=169$, we have $a^{2}+b^{2}=c^{2}$.

Answer:

By the Pythagorean Theorem, since $5^{2}+12^{2}=13^{2}$, the triangle is a right - triangle.