Question

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

2, 3, 4

Short answer

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

Step-by-step solution

Step 1. Given. 

The length of a triangle is2, 3, 4

Step 2. 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.

Step 3. 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 2, 3, and 4.

The longest side of the triangle is 4.

Therefore, the value c is 4.

Therefore, the values of a and b are 2 and 3 respectively.

Now, it can be obtained that:

$\begin{array}{c}{a}^2=2^2=4\\ b^2=3^2=9\\ c^2=4^2=16\\ a^2+b^2=4+9=13\end{array}$

It can be noticed that:

$\begin{array}{c}{a}^2+b^2=13\\ a^2+b^2\ne c^2\end{array}$

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.