Question

May I please receive help?

Answer 1

Problem 1 (on the left)

Answers:

$10^2+12^2 = \boldsymbol{244}\\\\14^2 = \boldsymbol{196}\\\\\text{What kind of triangle?} \textbf{ Acute}$

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Explanation:

The converse of the pythagorean theorem can be used to determine what kind of triangle we're dealing with.

Here are 3 important rules to have in your notes:

$\cdot \text{If } a^2+b^2 > c^2 \text{ then the triangle is } \underline{\text{acute}}\\\\\ \cdot \text{If } a^2+b^2 < c^2 \text{ then the triangle is } \underline{\text{obtuse}}\\\\\cdot \text{If } a^2+b^2 = c^2 \text{ then the triangle is a } \underline{\text{right}} \text{ triangle}$

The third rule is probably the most familiar as it's the pythagorean theorem itself.

In this case, we have $a^2+b^2 = 10^2+12^2 = 244$ larger than $c^2 = 14^2 = 196$, which fits the first rule mentioned.

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Problem 2 (on the right)

Answers:

$8^2+15^2 = \boldsymbol{289}\\\\17^2 = \boldsymbol{289}\\\\\text{What kind of triangle?} \textbf{ Right}$

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Explanation:

We follow the same idea as problem 1. This time, both $a^2+b^2$ and $c^2$ result in the same value (289). Therefore, $a^2+b^2 = c^2$ is a true equation, and we go for the third rule mentioned earlier. An 8-15-17 right triangle is one of the infinitely many pythagorean triples.