Question

which type of triangle satisfies the condition $a^{2}+b^{2}=c^{2}$?
a. obtuse triangle
b. acute triangle
c. right triangle
d. equilateral triangle
what type of triangle is formed if the sides are $a = 9, b = 12$, and $c = 15$?
a. right triangle
b. scalene triangle
c. obtuse triangle
d. acute triangle

Explanation:

Step1: Recall Pythagorean theorem

The Pythagorean theorem states that for a right - triangle, the sum of the squares of the two legs ($a$ and $b$) is equal to the square of the hypotenuse ($c$), i.e., $a^{2}+b^{2}=c^{2}$.

Step2: Answer first question

Since the condition $a^{2}+b^{2}=c^{2}$ is the Pythagorean theorem, the triangle is a right - triangle.

Step3: Check the second - triangle

Calculate $a^{2}+b^{2}$ and $c^{2}$ for $a = 9$, $b = 12$, and $c = 15$.
$a^{2}=9^{2}=81$, $b^{2}=12^{2}=144$, so $a^{2}+b^{2}=81 + 144=225$. And $c^{2}=15^{2}=225$. Since $a^{2}+b^{2}=c^{2}$, it is a right - triangle. Also, since $a
eq b
eq c$, it is a scalene triangle. But the most specific classification based on the Pythagorean - related property is a right - triangle.

Answer:

1. c. Right triangle
2. a. Right triangle