Question

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

5, 12, 13

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 the triangle is$5,12,13$.

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,$a$ and$b$ 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 5, 12, and 13.

The longest side of the triangle is 13.

Therefore, the value $c$is 13.

Therefore, the values of $a$and $b$are 5 and 12 respectively.

Now, it can be obtained that:

$\begin{array}{l}{a}^2=5^2=25\\ b^2={12}^2=144\\ c^2={13}^2=169\\ a^2+b^2=25+144=169\end{array}$

It can be noticed that:

$\begin{array}{l}{a}^2+b^2=169\\ a^2+b^2=c^2\end{array}$

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.