Question

find the perimeter of each polygon. assume that lines which appear to be tangent are tangent.
13)
14)

Explanation:

Problem 13)

Step1: Use tangent segment theorem

For a quadrilateral with an incircle, the sums of the lengths of opposite sides are equal. Let the unknown tangent segments be $x, y, z, w$. We know that tangent segments from a single external point to a circle are equal. So:

• The side of length 10.1 has an adjacent tangent segment of 3.5, so the corresponding tangent on the opposite side is also 3.5.
• The side of length 10.4 will pair with a tangent segment equal to $10.1 - 3.5 = 6.6$
• The side of length 8 will pair with a tangent segment equal to $10.4 - 6.6 = 3.8$
• The final unknown side segment is $8 - 3.8 = 4.2$

Step2: Calculate total perimeter

Sum all side lengths, using the property that $AB + CD = BC + AD$ for a tangential quadrilateral.
Perimeter $= 2\times(10.1 + 10.4) = 2\times20.5 = 41$
Or verify by summing all segments: $10.1 + 10.4 + 8 + (3.5 + 3.8) = 10.1+10.4+8+7.3=41$

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Problem 14)

Step1: Use tangent segment theorem

For a triangle with an incircle, tangent segments from a single external point are equal. The side of length 22.4 has a segment of 10.4, so the other tangent segment from that vertex is $22.4 - 10.4 = 12$.

Step2: Find missing side segments

The tangent segment on the side of length 18 that corresponds to the 12 segment is also 12, so the remaining segment on the 18 side is $18 - 12 = 6$. The final unknown side of the triangle is $10.4 + 6 = 16.4$.

Step3: Calculate total perimeter

Sum all three sides:
Perimeter $= 22.4 + 18 + 16.4$

Answer:

1. $\boldsymbol{41}$
2. $\boldsymbol{56.8}$