We are asked to find the perimeter of the polygon in the first picture of question 19. We are given a polygon circumscribed about a circle. The sides of the polygon are tangent to the circle. The given lengths of the tangent segments are 2, 3, 8, and 12.

1. Problem Description

We are asked to find the perimeter of the polygon in the first picture of question

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9. We are given a polygon circumscribed about a circle. The sides of the polygon are tangent to the circle. The given lengths of the tangent segments are 2, 3, 8, and

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2.

2. Solution Steps

Since tangent segments from the same exterior point are congruent, we can label the missing lengths of the sides.

Let the polygon have vertices A, B, C, and D in counterclockwise order.

Let the tangent point on side AB have length 12 from A. This means the remaining segment from the tangent point to B also has length

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2. Let the tangent point on side BC have length 3 from C. This means the remaining segment from the tangent point to B also has length

3. Let the tangent point on side CD have length 8 from C. This means the remaining segment from the tangent point to D also has length

8. Let the tangent point on side DA have length 2 from D. This means the remaining segment from the tangent point to A also has length

2. So, the length of the sides are:

$AB = 12 + 12 = 24$

$BC = 3 + 8 = 11$

$CD = 8 + 2 = 10$

$DA = 2 + 12 = 14$

The perimeter is the sum of the side lengths.

$P = AB + BC + CD + DA = 24 + 11 + 10 + 14$

$P = 59$

3. Final Answer

59