three circles are manually tangent externally. their centers form a triangle whose sides are of length 8, 9, 13. find the total area of the three circles

If the radii of the circles are x,y,z then each side of the triangle is the sum of two of the radii. So,

x+y=8
x+z=9
y+z=13
(x,y,z) = (2,6,7)

Since the area of a circle is πr^2, the area of all of the circles is
π(2^2+6^2+7^2) = 89π

To find the total area of the three circles, we need to determine the radius of each circle.

Let's denote the sides of the triangle formed by the centers of the circles as a, b, and c. In this case, the sides of the triangle are given as 8, 9, and 13.

Now, there is a known relationship between the sides of a triangle and the radius of a circle that is tangent to all three sides. This relationship is known as the "triangle inequality."

According to the triangle inequality, if a circle is tangent to all three sides of a triangle, the radius of the circle can be calculated using the following formula:

radius = (sqrt((s - a) * (s - b) * (s - c))) / s

where s is the semi-perimeter of the triangle, given by (a + b + c) / 2.

Let's calculate the semi-perimeter first:

s = (8 + 9 + 13) / 2
s = 30 / 2
s = 15

Now, we can substitute this value into the formula for the radius:

radius = (sqrt((15 - 8) * (15 - 9) * (15 - 13))) / 15

Simplifying further:

radius = (sqrt(7 * 6 * 2)) / 15
radius = (sqrt(84)) / 15
radius ≈ 9.165

Therefore, the radius of each circle is approximately 9.165.

Now, to find the area of a circle, we can use the formula:

area = π * radius^2

Let's calculate the area of one circle:

area = π * (9.165)^2
area ≈ 264.66

Since we have three circles, the total area is:

total area = 3 * 264.66
total area ≈ 793.98

Therefore, the total area of the three circles is approximately 793.98 square units.