Question

if $\angle prq \cong \angle prs$, $pq = 2y$, and $ps = y + 21$, what is $pq$?
$pq = \square$

Explanation:

Step1: Use the Angle - Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. In this case, since \(\angle PRQ\cong\angle PRS\), by the Angle - Bisector Theorem, \(PQ = PS\).

Step2: Set up the equation

Given \(PQ = 2y\) and \(PS=y + 21\), we set \(2y=y + 21\).
Subtract \(y\) from both sides: \(2y-y=y + 21-y\).
So, \(y = 21\).

Step3: Find the value of \(PS\)

Substitute \(y = 21\) into the expression for \(PS\). Since \(PS=y + 21\), then \(PS=21+21\).

Answer:

\(42\)