Question

based on the areas of the squares, determine whether the triangle shown is a right triangle.
b) how long is each side of the triangle?
a² = 58
c² = 64
b² = 10

Explanation:

Step1: Recall the relationship between square - area and side - length

The area of a square with side - length $s$ is $A = s^{2}$, so $s=\sqrt{A}$.

Step2: Calculate the side - lengths of the squares

For the square with area $a^{2}=58$, the side - length $a=\sqrt{58}$.
For the square with area $b^{2}=10$, the side - length $b = \sqrt{10}$.
For the square with area $c^{2}=64$, the side - length $c=\sqrt{64}=8$.

Step3: Check the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle, $a^{2}+b^{2}=c^{2}$.
Substitute the values: $a^{2}=58$, $b^{2}=10$, and $c^{2}=64$.
Calculate $a^{2}+b^{2}=58 + 10=68$.
Since $68
eq64$ (i.e., $a^{2}+b^{2}
eq c^{2}$), the triangle is not a right - triangle.

Answer:

The side - lengths of the triangle are $a=\sqrt{58}$, $b=\sqrt{10}$, and $c = 8$. The triangle is not a right - triangle.