Question

find the perimeter of δqrs.
q
3t - 1 4
o → p
8t - 1
12
r → 18 s
© algebra einstein, 2023

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \(OP\parallel RS\) in \(\triangle QRS\), we have \(\frac{QO}{QR}=\frac{QP}{QS}\).
\(QR=(3t - 1)+(8t - 1)=11t-2\), \(QS = 4 + 12=16\), \(QO = 3t - 1\), \(QP = 4\).
So, \(\frac{3t - 1}{11t-2}=\frac{4}{16}\).

Step2: Solve the proportion for \(t\)

Cross - multiply: \(16(3t - 1)=4(11t - 2)\).
Expand: \(48t-16 = 44t-8\).
Subtract \(44t\) from both sides: \(48t-44t-16=44t - 44t-8\), \(4t-16=-8\).
Add \(16\) to both sides: \(4t-16 + 16=-8 + 16\), \(4t=8\).
Divide by \(4\): \(t = 2\).

Step3: Find the lengths of the sides of \(\triangle QRS\)

\(QR=11t-2\), substitute \(t = 2\): \(QR=11\times2-2=20\).
\(QS=16\), \(RS = 18\).

Step4: Calculate the perimeter \(P\) of \(\triangle QRS\)

\(P=QR + QS+RS\).
\(P=20 + 16+18\).

Answer:

\(54\)