Chapter 14, Problem 23CT

Solution

To find: the remaining sides and angles of the given triangle.

  $D={65.7}^{\circ },E={24.3}^{\circ },d\approx 46.5$ .

Given:

  $e=21,f=51$ .

Concept used:

The sum of two acute angles of a right triangle is complementary.

Pythagorean Theorem: In any right triangle, the sum of squares of the lengths of two legs is equal to the

Square of the hypotenuse.

Calculation:

Let us draw the right triangle by using the given values as follows:

  

In the above triangle,

  $\begin{array}{c}\sin E=\frac{21}{51}\\ E={24.31}^{\circ }\\ E\approx {24.3}^{\circ }\end{array}$

Now since the sum of two acute angles of a right triangle is complementary, therefore

  $\begin{array}{c}D+E={90}^{\circ }\\ D+{24.3}^{\circ }={90}^{\circ }\\ D={65.7}^{\circ }\end{array}$

Now by using the Pythagorean Theorem in the triangle ABC, it follows that

  $\begin{array}{c}{21}^2+d^2={51}^2\\ 441+d^2=2601\\ d^2=2160\\ d=\sqrt{2160}\\ d=46.47\\ d\approx 46.5\end{array}$