Question

find the perimeter of the polygon if $m\angle a = m\angle c$.
$p = ?$ ft
all the segments are tangent to circle o.

Explanation:

Step1: Use tangent segment theorem

From a single external point, tangent segments to a circle are equal in length. So:

• From point A: $AB_{tangent}=4$ ft, so the other tangent from A to the circle is also 4 ft.
• From point D: $AD_{tangent}=9$ ft, so the other tangent from D to the circle is also 9 ft.
• From point B: $BA_{tangent}=4$ ft, so the other tangent from B to the circle is 6 ft.
• Since $m\angle A = m\angle C$, the tangent segments from point C mirror those from point A: the tangent segments from C are 9 ft and 6 ft, matching the lengths from D and B respectively.

Step2: Identify all side lengths

• $AD = 9 + 4 = 13$ ft
• $AB = 4 + 6 = 10$ ft
• $BC = 6 + 9 = 15$ ft
• $CD = 9 + 6 = 15$ ft

Step3: Calculate perimeter

Sum all side lengths: $P = AD + AB + BC + CD$
$P = 13 + 10 + 15 + 15$

Answer:

43 ft