Chapter 5.1, Problem 23WE

To determine

To find: the values of x and y for a given parallelogram.

Answer to Problem 23WE

The value of x is $5$

The value of y is $4$

Explanation of Solution

Given information:

A diagram representing a parallelogram with its diagonals and measure of sides being given.

Formula used:

1) By theorem 5-3, the diagonals of a parallelogram divide themselves into two equal segments about their point of intersection.

2) By Theorem 5-1 ,the opposite sides of a parallelogram are in congruence which imply they have the same measure

Calculation:

Consider the figure given below.

  

Based on the Theorem 5-3, it can be said that the two segments of each diagonal are equal to one another. In the case of the given parallelogram, the values of diagonals are given and the theorem can be expressed as,

  $2x+2y=18$......(1)

Now, based on Theorem 5-1, the opposite sides can be expressed as,

  $2x+5y=30$......(2)

Based on the above two equations, the values of x and y can be calculated as follows. Subtracting the (1) from (2) gives,

  $\begin{array}{c}5y-2y=30-18\\ 3y=12\end{array}$

Dividing both sides by 3, the value of y can be obtained as,

  $\begin{array}{c}y=\frac{12}{3}\\ =4\end{array}$

Substituting the value of y in (1), x can be obtained as,

  $\begin{array}{c}2x+2\left(4\right)=18\\ 2x=18-8\\ x=\frac{10}{2}\\ =5\end{array}$