Chapter 6, Problem 36RE

To determine

To Find: The lengths of the diagonals of a parallelogram are 30 meters and 40 meters. Find the lengths of the sides of the parallelogram when the diagonals intersect at an angle of ${34}^{°}$ .

Answer to Problem 36RE

Side of parallelogramare $y=33.51$ , and $x=11.27$

Explanation of Solution

Given information:

Length of Diagonal of parallelogram are given as 30 meters and 40 meters and the angle of intersection is given as ${34}^{°}$

Formula used:

Sine Rule:

  $\cos C=\frac{a^2+b^2-c^2}{2(a)(b)}$

Calculation:

As we know diagonal of parallelogram bisect each other.

  

Therefore $AO=15$ , $BO=20$

Now applying Cosine rule on the triangle $AOB$ :

  $\begin{array}{l}\cos C=\frac{a^2+b^2-x^2}{2(a)(b)}\\ \cos {34}^{°}=\frac{{15}^2+{20}^2-x^{°}}{2(20)(15)}\\ 0.83=\frac{225+400-x^2}{600}\\ 498=625-x^2\end{array}$

  $\begin{array}{l}{x}^2=127\\ \boxed{x=11.27}\end{array}$

Now applying Cosine rule on the triangle $AOD$ :

  $\begin{array}{l}\cos C=\frac{a^2+b^2-y^2}{2(a)(b)}\\ \cos {146}^{°}=\frac{{15}^2+{20}^2-y^{°}}{2(20)(15)}\\ -0.83=\frac{225+400-y^2}{600}\\ -0.83x600=625-y^2\end{array}$

  $\begin{array}{l}{y}^2=1123\\ \boxed{y=33.51}\end{array}$