Question

if (overline{ps}congoverline{rs}), (pq = 4a), and (qr=a + 27), what is the value of (a)?(a=)

Explanation:

Step1: Use the property of perpendicular bisector

Since \( \overline{PS}\cong\overline{RS}\) and \(QS\perp PR\), by the perpendicular bisector theorem, \(PQ = QR\).

Step2: Set up the equation

We know that \(PQ = 4a\) and \(QR=a + 27\). Substituting into \(PQ=QR\), we get the equation \(4a=a + 27\).

Step3: Solve the equation for \(a\)

Subtract \(a\) from both sides of the equation \(4a=a + 27\):
\(4a-a=a + 27-a\)
\(3a=27\)
Divide both sides by \(3\):
\(a=\frac{27}{3}\)

Answer:

\(9\)