Video Transcript
Three tangential circles have radii
200 centimeters, 120 centimeters, and 110 centimeters. Find the area of the triangle
formed by connecting the centers of the circles. Give your answer to two decimal
places.
We’re given the radii of three
tangential circles. So let’s begin by sketching the
circles. We want to find the area of the
triangle formed by connecting the centers of the circles. And to find the area of this
triangle, we can use Heron’s formula. This says that for a triangle with
side lengths 𝑎, 𝑏, and 𝑐, the area is equal to the square root of 𝑠 times 𝑠
minus 𝑎 times 𝑠 minus 𝑏 times 𝑠 minus 𝑐, where 𝑠 is the semiperimeter. That’s half the sum of the side
lengths.
Our first step now is to work out
the side lengths of our triangle. And these are each the sum of the
radii of two circles. So, for example, calling our first
side length 𝑎, this is the sum of the radius of the largest circle, 200
centimeters, and that of the second largest circle, 120 centimeters, which is 320
centimeters. Similarly, our next side 𝑏 is the
sum of the largest radius, 200 centimeters, with the smallest radius, 110
centimeters. And that’s 310 centimeters. Our third side length 𝑐 of the
triangle is the sum of the two smaller radii. That’s 120 plus 110 centimeters,
which is 230 centimeters.
So, with our side lengths 320, 310,
and 230 centimeters, we can now work out the semiperimeter 𝑠 to use in Heron’s
formula. That’s 320, which is 𝑎, plus 310,
which is 𝑏, plus 230, which is 𝑐, all over two. And that’s 860 over two, which is
430. So 𝑠 is equal to 430.
So now making some space, we can
use the values we’ve calculated in Heron’s formula to find the area of the
triangle. So we have the area 𝑎 is equal to
the square root of 430 times 110 times 120 times 200. That’s the positive square root,
since area is always positive, of 1,135,200,000, which is approximately equal to
33,692.72919 and which is 33,692.73 to two decimal places.
Hence, the area of the triangle
formed by connecting the centers of the three tangential circles, to two decimal
places, is 33,692.73 centimeters squared.
