Question

if $overline{pq}congoverline{qr}$, $ps = c + 20$, and $rs = 3c$, what is $pr? pr=square$

Explanation:

Step1: Use property of perpendicular bisector

Since $\overline{PQ}\cong\overline{QR}$ and $QS$ is perpendicular to $PR$, by the property of the perpendicular - bisector of a line segment (a point equidistant from the endpoints of a line segment lies on the perpendicular bisector of the line segment, and conversely, the perpendicular bisector of a line segment consists of all points that are equidistant from the endpoints of the line segment), $QS$ is the perpendicular bisector of $\overline{PR}$. So, $PS = RS$.
$c + 20=3c$

Step2: Solve for $c$

Subtract $c$ from both sides of the equation $c + 20=3c$.
$20=3c - c$
$20 = 2c$
Divide both sides by 2: $c=\frac{20}{2}=10$

Step3: Find $PR$

Since $PR=PS + RS$ and $PS = RS$, and $PS=c + 20$, $RS = 3c$. Substitute $c = 10$ into the expressions for $PS$ and $RS$.
$PS=10 + 20=30$, $RS = 3\times10 = 30$
$PR=PS+RS=30 + 30=60$

Answer:

$60$