Question

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

15, 19, 23

Short answer

The given set of measures is not the lengths of the sides of a right triangle.

Step-by-step solution

Step1. Given 

The length of a triangle is$15,19,23$

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 15, 19, and 23.

The longest side of the triangle is 23.

Therefore, the value$c$ is 23.

Therefore, the values of$a$ and$b$ are 15 and 19 respectively.

Now, it can be obtained that:

$\begin{array}{l}{a}^2={15}^2=225\\ b^2={19}^2=361\\ c^2={23}^2=529\\ a^2+b^2=225+361=586\end{array}$

It can be noticed that:

role="math" localid="1648103513440" $\begin{array}{l}{a}^2+b^2=586\\ a^2+b^2\ne c^2\end{array}$

As, $c^2\ne a^2+b^2$, therefore, the triangle is not a right triangle.

Therefore, the given set of measures is not the lengths of the sides of a right triangle.