Answer:
Since PQRS is a rectangle and the area of PQRS is 84 cm², then:
PQ x PS = 84
Since PQRS is a rectangle, PS = QR, so we can substitute PS for QR:
PQ x QR = 84
We also know that XY is parallel to PS, so triangle RXY is similar to triangle RQS:
RY/QS = XY/PS
9/QS = XY/PS
QS = 9XY/PS
The area of PXYS is 21 cm², so:
XY x PS = 21
Substitute QS for 9XY/PS:
XY x (9XY/QS) = 21
Simplify:
9XY²/QS = 21
XY² = 21QS/9
Substitute QS for PQ, since PQ = QS:
XY² = 21PQ/9
Substitute 84/PQ for PQ in the above equation:
XY² = 21(84/PQ)/9
XY² = 196/PQ
Now we can substitute PQ for 84/XY in the equation PQ x QR = 84:
(84/XY) x QR = 84
QR = XY
Substitute QR for PS in the equation XY x PS = 21:
XY² = 21/XY
Multiply both sides by XY:
XY³ = 21
XY = ∛21
Since RY = 9 cm, we can find QS:
9/QS = ∛21/PS
QS = 9PS/∛21
Now we can substitute QS and XY into the equation XY x PS = 21 and solve for PS:
(∛21) x PS = 21/((9/∛21))
PS = (21/((9/∛21)))/∛21
PS = 7∛21
Finally, we can solve for a and b:
a = PQ - XY = (84/PQ) - (∛21)
Substitute PQ for 84/XY:
a = (XY²/84) - ∛21
Substitute XY for ∛21:
a = (∛21²/84) - ∛21
a = 1/4 - ∛21
b = RY - QS = 9 - (9PS/∛21)
Substitute PS for 7∛21:
b = 9 - (9(7∛21)/∛21)
b = -54/∛21 + 9∛21
Explanation: