Question

circle c is inscribed in triangle qsu. what is the perimeter of triangle qsu? 3 units 16 units 30 units 40 units

Explanation:

Step1: Recall tangent - segment property

If two - tangent segments are drawn from an external point to a circle, then they are congruent.
So, $QA = QR = 10$, $CA=CT = 4$, and $ST=SR$.
Since $SR = 2x$ and $ST=x + 3$, we have $2x=x + 3$.

Step2: Solve the equation for $x$

Subtract $x$ from both sides of the equation $2x=x + 3$.
$2x−x=x + 3−x$, which gives $x = 3$.

Step3: Find the lengths of the sides of the triangle

$QS=QR+RS=10 + 2x$, substituting $x = 3$, we get $QS=10+6 = 16$.
$SU=ST+TU=(x + 3)+4$, substituting $x = 3$, we get $SU=6 + 4=10$.
$QU=QA+AU=10 + 4=14$.

Step4: Calculate the perimeter of $\triangle QSU$

The perimeter $P$ of $\triangle QSU$ is $P=QS+SU+QU$.
$P=16 + 10+14=40$ units.

Answer:

40 units