Question

1. find the perimeter of the triangle circumscribed about the circle below. (options: 45 cm, 50 cm, 63 cm, 68 cm)

Explanation:

Step1: Recall tangent segment property

For a triangle circumscribed about a circle, the lengths of two tangent segments from a single external point to the circle are equal.

Step2: Identify equal tangent lengths

Let the unknown tangent segments be $x$ (from the bottom vertex to the tangent on the 29 cm side), $y$ (from the top-right vertex to the tangent on the 29 cm side), and $z$ (from the top-left vertex to the tangent on the vertical side). We know:

• From top-left: $z = 11$ cm, so the other tangent from this vertex is also 11 cm.
• From top-right: $z' = 5$ cm, so the other tangent from this vertex is also 5 cm.
• The long side: $11 + 5 + x + y = 29$? No, correct: the long side is $x + y = 29$, where $x$ is the tangent from bottom to top-left side, $y$ is tangent from bottom to top-right side.

Step3: Calculate total perimeter

The perimeter $P$ is $2\times(11 + 5 + x + y)$? No, simplify: $P = 2\times(11 + 5) + 2\times29$? No, correct: $P = 2\times(11 + 5 + (29 - 11 - 5))$? Wait, simpler: Perimeter = $2\times(11 + 5 + 29 - 11 - 5 + 11 + 5)$? No, use the rule: For a tangential triangle, perimeter = $2\times$(sum of one set of tangent segments from each vertex). We have two known tangent segments: 11 cm and 5 cm, and the third side's total length is 29 cm, which is equal to $11 + 5$? No, the correct way: Let the three pairs of equal tangents be $a=11$ cm, $b=5$ cm, $c$. Then $a + c = 29$, so $c = 29 - 11 = 18$ cm. Then perimeter is $2(a + b + c) = 2(11 + 5 + 18)$

Step4: Compute the value

$2(11 + 5 + 18) = 2(34) = 68$ cm

Answer:

68 cm