the 3 circles in the adjoining figure are externally tangent to each other, and each side of the triangle is tangent to two of the circles. if each circle has a radus of 2, then find the perimeter of the triangle.

diagram: https://latex.artofproblemsolving.com/0/9/e/09eb9cd5e511702149aa9236e78030f2f0300893.png

Guest Mar 16, 2020

#1**0 **

Find the centers of the bottom-left circle and the bottom-right circles.

Draw a line segment that connects these two points. This line segment will go through the common point of tangency of the circles.

Since this is the length of two radii, it will have a length of 4.

Now, draw line segments from these two centers perpendicular to the base side of the triangle.

The distance between the points of intersection where these two radii hit the base will be 4.

Now, draw the line segment from the center of the bottom-left circle to the bottom-left corner of the triangle.

This line segement, along with the perpendicular radius drawn to the bottom side of the triangle, will help determine a right triangle.

Since the bottom-left corner of the triangle has 60^{o}, the triangle is a 30^{o} - 60^{o} - 90^{o} triangle whose short side has value 2.

Therefore, the length along the bottom side of this triangle is 2·sqrt(2).

From this, you can determine the perimeter of the triangle.

geno3141 Mar 16, 2020