Question

using the two tangent theorem
circle o is inscribed in the given triangle.
what is the perimeter of the triangle?
22 units
30 units
44 units
60 units

Explanation:

Step1: Assign tangent segment variables

Let the tangent segments from each vertex be:

• Left vertex: segment length $12$ (to $P$) and $x$ (to $Q$)
• Right vertex: segment length $6$ (to $Q$) and $y$ (to $R$)
• Bottom vertex: segment length $4$ (to $P$) and $4$ (to $R$, by Two Tangent Theorem)

Step2: Find missing segment lengths

By Two Tangent Theorem:
$x = 12$, $y = 6$

Step3: Calculate side lengths

• Left side: $12 + 4 = 16$
• Right side: $6 + 4 = 10$
• Top side: $12 + 6 = 18$

Step4: Compute perimeter

Add all side lengths:
$$16 + 10 + 18 = 44$$

Answer:

44 units