Question

1. if ( overline{qt} ) is the perpendicular bisector of ( overline{pr} ), find each measure.

( x=)
( y=)
( pq=)
( qr=)
( ps=)
( sr=)
( pt=)
( pr=)

Explanation:

Step1: Find \(x\) using \(PS = SR\)

Since \(PS=4x + 4\) and \(SR = 7x-17\), and \(PS=SR\) (by the property of perpendicular bisector for segments \(PS\) and \(SR\) related to point \(S\) on the perpendicular bisector of \(PR\)), we have the equation \(4x + 4=7x-17\).
Subtract \(4x\) from both sides: \(4=3x - 17\).
Add \(17\) to both sides: \(3x=4 + 17=21\).
Divide by \(3\): \(x=\frac{21}{3}=7\).

Step2: Find \(y\) using \(PQ = QR\)

Since \(PQ = 5y-31\) and \(QR=2y + 5\), and \(PQ = QR\) (by the property of perpendicular bisector for segments \(PQ\) and \(QR\) related to point \(Q\) on the perpendicular bisector of \(PR\)), we have the equation \(5y-31=2y + 5\).
Subtract \(2y\) from both sides: \(3y-31 = 5\).
Add \(31\) to both sides: \(3y=5 + 31=36\).
Divide by \(3\): \(y=\frac{36}{3}=12\).

Step3: Calculate \(PQ\)

Substitute \(y = 12\) into \(PQ=5y-31\).
\(PQ=5\times12-31=60 - 31=29\).

Step4: Calculate \(QR\)

Substitute \(y = 12\) into \(QR=2y + 5\).
\(QR=2\times12+5=24 + 5=29\).

Step5: Calculate \(PS\)

Substitute \(x = 7\) into \(PS=4x + 4\).
\(PS=4\times7+4=28 + 4=32\).

Step6: Calculate \(SR\)

Substitute \(x = 7\) into \(SR=7x-17\).
\(SR=7\times7-17=49 - 17=32\).

Step7: Calculate \(PT\)

Since \(PT=\frac{1}{2}PR\) and \(PR=PS + SR\). First, \(PR=32+32 = 64\). Then \(PT=\frac{64}{2}=32\).

Step8: Calculate \(PR\)

\(PR=PS + SR\), with \(PS = 32\) and \(SR=32\), so \(PR=32+32=64\).

Answer:

\(x = 7\)
\(y = 12\)
\(PQ=29\)
\(QR = 29\)
\(PS=32\)
\(SR=32\)
\(PT=32\)
\(PR=64\)