Question

What values must x and y have to make the quadrilateral a parallelogram?

Short answer

The values that must x and y must have to make the quadrilateral a parallelogram are 20 and 6 respectively.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Find the value of x.

As the diagonals of the parallelogram bisect each other.

Therefore, to make the given quadrilateral a parallelogram, the diagonals of the parallelogram must bisect each other.

Therefore, the value of x can be obtained as:

$\begin{array}{c}3x-40=x\\ 3x-x=40\\ 2x=40\\ x=\frac{40}{2}\\ x=20\end{array}$

Therefore, the value of x is 20.

Step 3. Find the value of y.

As the diagonals of the parallelogram bisect each other.

Therefore, to make the given quadrilateral a parallelogram, the diagonals of the parallelogram must bisect each other.

Therefore, the value of y can be obtained as:

$\begin{array}{c}{y}^2=y+30\\ y^2-y-30=0\\ y^2-6y+5y-30=0\\ y\left(y-6\right)+5\left(y-6\right)=0\\ \left(y-6\right)\left(y+5\right)=0\\ y=6,-5\end{array}$

As the length of any side cannot be negative, therefore the only possible value of y is 6.

Therefore, the value of y is 6.