For what values of y and x must ABCD and PQRS be a parallelogram?

Question

P 3x - 5; x + 2 S 2x + 1 R

Answer 1

To determine the values of y and x that make ABCD and PQRS a parallelogram, we need to equate the opposite sides and opposite angles of the two quadrilaterals.

In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure.

For ABCD:
AB = CD
AD = BC

For PQRS:
PQ = SR
PS = QR

To determine the values of y and x, we can equate the corresponding sides of the two quadrilaterals and solve for y and x.

AB = PQ
CD = SR
AD = PS
BC = QR

Since P = 3x - 5, Q = x + 2, S = 2x + 1, and R = y:

3x - 5 = x + 2
2x + 1 = y

Simplifying the equations:

2x = 7
x = 7/2

Substituting the value of x into the equation 2x + 1 = y:

2(7/2) + 1 = y
7 + 1 = y
y = 8

Therefore, for ABCD and PQRS to be a parallelogram, y must be equal to 8 and x must be equal to 7/2.