all sides of quadrilateral abcd are tangent to circle p. what is the perimeter of the quadrilateral below? units (image of quadrilateral abcd with circle p inside, side lengths: ab segment with 12, ad segment with 4, dc segment with 9, bc segment with 4 markings)

Question

all sides of quadrilateral abcd are tangent to circle p. what is the perimeter of the quadrilateral below?   units (image of quadrilateral abcd with circle p inside, side lengths: ab segment with 12, ad segment with 4, dc segment with 9, bc segment with 4 markings)

Accepted Answer

# Explanation:
## Step1: Recall Tangent Segments Property
For a quadrilateral tangent to a circle (a tangential quadrilateral), the lengths of the two tangent segments from a single external point to the circle are equal. So, if we denote the tangent segments: from $ A $, the two tangents are equal; from $ B $, the two tangents are equal; from $ C $, the two tangents are equal; from $ D $, the two tangents are equal.

Let's label the tangent points. Let the tangent from $ A $ to the circle have length $ 4 $ (given), so the other tangent from $ A $ (to the adjacent side) is also $ 4 $. The tangent from $ B $ along $ AB $ is $ 12 $, so the other tangent from $ B $ (to the side $ BC $) is equal to the tangent from $ A $ along $ AB $? Wait, no, let's list the sides:

- Let the sides be $ AB $, $ BC $, $ CD $, $ DA $.
- From point $ A $: the two tangent segments (on $ AD $ and $ AB $): one is $ 4 $ (on $ AD $), so the other (on $ AB $) is also $ 4 $? Wait, no, looking at the diagram: $ AD $ has a tangent segment of length $ 4 $ (from $ A $ to the tangent point on $ AD $), and $ AB $ has a tangent segment of length $ 12 $ (from $ A $ to the tangent point on $ AB $)? Wait, no, the standard property is that for a tangential quadrilateral, the sum of the lengths of opposite sides are equal. Wait, actually, the perimeter of a tangential quadrilateral is $ 2 \times (sum of two adjacent tangent - segment sums) $? Wait, no, the correct property is that in a tangential quadrilateral, $ AB + CD = BC + AD $, and the perimeter is $ 2(AB + CD) $ or $ 2(BC + AD) $.

Wait, let's identify the tangent lengths:

- From $ A $: the two tangent segments (to the circle) are equal. Let's say the tangent on $ AD $ is $ 4 $, so the tangent on $ AB $ from $ A $ is also $ 4 $.
- From $ B $: the tangent on $ AB $ (from $ B $) is $ 12 - 4 = 8 $? Wait, no, maybe better to use the property that in a tangential quadrilateral, the sum of the lengths of two opposite sides is equal to the sum of the lengths of the other two opposite sides. Wait, actually, the perimeter $ P = 2(a + b) $, where $ a $ and $ b $ are the sums of the tangent segments from two opposite vertices.

Wait, looking at the diagram:

- The tangent segment from $ A $ to the circle on $ AD $ is $ 4 $, so the tangent segment from $ A $ to the circle on $ AB $ is also $ 4 $.
- The tangent segment from $ B $ to the circle on $ AB $ is $ 12 - 4 = 8 $? No, wait, the length of $ AB $ is the sum of the two tangent segments from $ A $ and $ B $ to the circle on $ AB $. Wait, no, the length of $ AB $ is the sum of the tangent from $ A $ to the circle on $ AB $ and the tangent from $ B $ to the circle on $ AB $. Similarly, $ BC $ is the sum of the tangent from $ B $ to the circle on $ BC $ and the tangent from $ C $ to the circle on $ BC $, $ CD $ is the sum of the tangent from $ C $ to the circle on $ CD $ and the tangent from $ D $ to the circle on $ CD $, and $ DA $ is the sum of the tangent from $ D $ to the circle on $ DA $ and the tangent from $ A $ to the circle on $ DA $.

But by the tangent -[SSE onError error]

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Explanation:

Step1: Recall Tangent Segments Property

Answer:

Step1: Recall Tangent Segments Property