Question

1. if \\( \triangle pqr \\) is an equilateral triangle, \\( pq = 18x + 1 \\), \\( qr = 24x - 17 \\), and \\( pr = 15x + 10 \\), find \\( x \\) and the measure of each side.

Explanation:

Step1: Set up the equation

Since \( \triangle PQR\) is equilateral, \(PQ = QR\).
\(18x + 1=24x - 17\)

Step2: Solve for \(x\)

Subtract \(18x\) from both sides:
\(1 = 6x-17\)
Add \(17\) to both sides:
\(18 = 6x\)
Divide both sides by \(6\):
\(x = 3\)

Step3: Find the length of each side

Substitute \(x = 3\) into \(PQ=18x + 1\):
\(PQ=18\times3 + 1=54 + 1=55\)
Check with \(QR = 24x-17\): \(24\times3-17=72 - 17=55\)
Check with \(PR=15x + 10\): \(15\times3+10=45 + 10=55\)

Answer:

\(x = 3\), and the measure of each side is \(55\)