Question

1. in parallelogram defg, dh = x + 5, hf = 2y, gh = 3x - 1, and he = 5y + 4. find the values of x and y.

x = 35, y = 20
x = 21, y = 20
x = 20, y = 35
x = 21, y = 37

Explanation:

Step1: Recall parallelogram property

In a parallelogram, the diagonals bisect each other. So, $DH = HF$ and $GH=HE$.

Step2: Set up equations from equal - length segments

We get the equations: $x + 5=2y$ and $3x−1 = 5y + 4$.
From $x + 5=2y$, we can express $x$ as $x=2y - 5$.

Step3: Substitute $x$ into the second equation

Substitute $x = 2y-5$ into $3x−1 = 5y + 4$.
$3(2y - 5)-1=5y + 4$.
Expand the left - hand side: $6y-15 - 1=5y + 4$.
$6y-16 = 5y + 4$.

Step4: Solve for $y$

Subtract $5y$ from both sides: $6y-5y-16=5y-5y + 4$.
$y-16 = 4$.
Add 16 to both sides: $y=4 + 16=20$.

Step5: Solve for $x$

Substitute $y = 20$ into $x=2y - 5$.
$x=2\times20-5$.
$x = 40-5=35$.

Answer:

$x = 35,y = 20$