Question

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

3, 4, 5

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$3,4,5$

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 3, 4, and 5.

The longest side of the triangle is 5.

Therefore, the value$c$ is 5.

Therefore, the values of $a$and $b$are 3 and 4 respectively.

Now, it can be obtained that:

$\begin{array}{l}{a}^2=3^2=9\\ b^2=4^2=16\\ c^2=5^2=25\\ a^2+b^2=9+16=25\end{array}$

It can be noticed that:

$\begin{array}{c}{a}^2+b^2=25\\ a^2+b^2=c^2\end{array}$

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.