Chapter 7.1, Problem 27WE

To determine

To calculate: The measure of each angle, if the ratio of the acute angles of a right triangle is $5:7$ .

Answer to Problem 27WE

The measure of the acute angles of a right triangle are $37.5°and52.5°$ .

Explanation of Solution

Given information:

The ratio of the angles of a triangle is $5:7$ .

Formula Used:

Property of right angled triangle:

One angle is of $90°$ and the sum of all the other two acute angles of a right-angled triangle is $90°$ .

Calculation:

Let $A,BandC$ be the angles of a triangle.

So, $B+C=90°$ .

Consider that the ratio of the acute angles of a right triangle is $5:7$ .

Let the angles be $5xand7x$ .

Now, find the value of $x$ .

  $\begin{array}{c}5x+7x=90°\\ 12x=90°\\ x=\frac{90°}{12}\\ x=\frac{15}{2}\end{array}$

Now, we find the angles:

Substitute $\frac{15}{2}$ for $x$ in the angles are $5xand7x$ .

  $\begin{array}{c}5x=5\times \frac{15}{2}\\ =\frac{75}{2}\\ =37.5°\end{array}$

  $\begin{array}{c}7x=7\times \frac{15}{2}\\ =\frac{105}{2}\\ =52.5°\end{array}$

Hence, the measure of the acute angles of the right triangle are $37.5°and52.5°$ .