Question

if $overline{rs}congoverline{st}$, $qr = 3p + 16$, and $qt = 8p - 99$, what is $qr$?
qr =
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Explanation:

Step1: Use property of perpendicular bisector

Since $\overline{RS}\cong\overline{ST}$ and $QS$ is perpendicular to $RT$, then $QS$ is the perpendicular - bisector of $RT$. By the property of the perpendicular bisector, $QR = QT$.
So, $3p + 16=8p - 99$.

Step2: Solve the equation for $p$

Subtract $3p$ from both sides of the equation:
$16 = 8p-3p - 99$, which simplifies to $16 = 5p-99$.
Add 99 to both sides: $16 + 99=5p$, so $115 = 5p$.
Divide both sides by 5: $p=\frac{115}{5}=23$.

Step3: Find the value of $QR$

Substitute $p = 23$ into the expression for $QR$.
$QR=3p + 16$.
$QR=3\times23+16$.
$QR = 69+16$.
$QR = 85$.

Answer:

$85$