Question

solve for the value of x.

question 14 6.25 pts

find qr.

Explanation:

Step1: Use the perpendicular bisector theorem

According to the perpendicular bisector theorem, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. In the first triangle, since \(CD\) is the perpendicular bisector of \(AB\), we have \(CB = CA\). So \(x + 5=15\).

Step2: Solve the equation for \(x\)

Subtract \(5\) from both sides of the equation \(x + 5=15\). Using the subtraction property of equality: \(x=15 - 5\).

Step3: Use the perpendicular bisector theorem for the second problem

In the second figure, since \(QS\) is the perpendicular bisector of \(PR\), then \(QP = QR\). So \(5x=3x + 8\).

Step4: Solve the equation \(5x=3x + 8\) for \(x\)

Subtract \(3x\) from both sides: \(5x-3x=3x + 8-3x\), which gives \(2x=8\). Then divide both sides by \(2\): \(x = 4\).

Step5: Find \(QR\)

Substitute \(x = 4\) into \(QR=3x + 8\). So \(QR=3\times4+8=12 + 8=20\).

Answer:

For the first problem: \(10\) (from \(x=15 - 5\)). For the second problem: \(20\) (from \(QR = 3x+8\) with \(x = 4\)).