Question

Determine whether each set of measures can be the lengths of the sides of a right triangle.

7, 24, 25

Short answer

The given set of measures is the lengths of the sides of a right triangle.

Step-by-step solution

Step1. Given 

The length of a triangle is$7,24,25$.

Step2. Determine how to identify whether the given measures are the sides of the right triangle, acute triangle an obtuse triangle.

If,$c$ is the longest side of the triangle, and are the other two sides of the triangle then:

(i) If $c^2<a^2+b^2$, then the triangle is acute.

(ii) If $c^2=a^2+b^2$, then the triangle is a right triangle.

(iii) If $c^2>a^2+b^2$, then the triangle is obtuse.

Step3. Determine whether the given set of measures can be the lengths of the sides of a right triangle.

The triangle is having sides of measures 7, 24, and 25.

The longest side of the triangle is 25.

Therefore, the value $c$is 25.

Therefore, the values oflocalid="1648102650851" $a$andlocalid="1648102658914" $b$are 7 and 24 respectively.

Now, it can be obtained that:

$\begin{array}{l}{a}^2=7^2=49\\ b^2={24}^2=576\\ c^2={25}^2=625\\ a^2+b^2=49+576=625\end{array}$

It can be noticed that:

localid="1648102595545" $\begin{array}{l}{a}^2+b^2=625\\ a^2+b^2=c^2\end{array}$

As, $c^2=a^2+b^2$, therefore, the triangle is a right triangle.

Therefore, the given set of measures is the lengths of the sides of a right triangle.