Answer :
The value of x is 8 and y is 30.
To find the values of x and y in triangle QPS, we'll utilize the properties of perpendicular bisectors and the midpoint theorem.
Given that PR is a perpendicular bisector, it divides QS into two equal segments because R is the midpoint of QS. So, QS is divided into RS and QR, each being half the length of QS.
Given:
[tex]\( RS = 7x + 27 \)\\ \( QR = 10x + 3 \)\\\( \angle PRS = 3y \)[/tex]
Using the properties of a perpendicular bisector, we know that RS = QR.
So, we equate the expressions for RS and QR:
7x + 27 = 10x + 3
Solving for x:
[tex]\[ 7x - 10x = 3 - 27 \]\[ -3x = -24 \]\[ x = 8 \][/tex]
Now, substituting x = 8 into either RS or QR, we find:
[tex]\[ RS = 7(8) + 27 = 56 + 27 = 83 \]\[ QR = 10(8) + 3 = 80 + 3 = 83 \][/tex]
Since RS = QR, PR is indeed a perpendicular bisector.
Now, to find y, we use the given [tex]\( \angle PRS = 3y \)[/tex]:
[tex]\[ 3y = 90 \]\[ y = 30 \][/tex]
Thus, x = 8 and y = 30.
