Question

1. in parallelogram defg, dh = x + 2, hf = 2y, gh = 3x - 3, and he = 5y + 1. find the values of x and y.

x = 11, y = 20

x = 18, y = 10

x = 20, y = 11

x = 10, y = 18

Explanation:

Step1: Recall parallelogram property

In a parallelogram, the diagonals bisect each other. So, \(DH = HF\) and \(GH=HE\).
We get the equations: \(x + 2=2y\) and \(3x-3 = 5y+1\).

Step2: Rearrange the first - equation

From \(x + 2=2y\), we can express \(x\) in terms of \(y\) as \(x=2y - 2\).

Step3: Substitute \(x\) into the second - equation

Substitute \(x = 2y-2\) into \(3x-3 = 5y + 1\).
\[

$$\begin{align*} 3(2y-2)-3&=5y + 1\\ 6y-6-3&=5y + 1\\ 6y-9&=5y + 1\\ 6y-5y&=1 + 9\\ y&=10 \end{align*}$$

\]

Step4: Find the value of \(x\)

Substitute \(y = 10\) into \(x=2y - 2\).
\(x=2\times10-2=18\)

Answer:

\(x = 18,y = 10\)