Four circles with centers A,B,C,and D are mutually tangent. The areas of circles A,B,C, and D are 25pi, 100pi, 16pi, and 225pi respectively. How many units are in the perimeter of quadrilateral ABCD?

The radii of the circles are 5, 10, 4 and 15.

If they can be arranged to be mutually tangent, the quadrilateral ABCD connects the centers of the circles, and has perimeter = 2(4 + 5 + 10 + 15) = 68

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the answer is 68

To find the perimeter of quadrilateral ABCD, we need to determine the length of the sides.

First, let's analyze the circles and try to find any patterns or relationships between their areas.

The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

Let's denote the radii of the circles with centers A, B, C, and D as rA, rB, rC, and rD, respectively.

Given that the areas of circles A, B, C, and D are 25π, 100π, 16π, and 225π, we can equate these values to their respective formulas:

πrA^2 = 25π
πrB^2 = 100π
πrC^2 = 16π
πrD^2 = 225π

Simplifying the equations by canceling out the common factor of π, we have:

rA^2 = 25
rB^2 = 100
rC^2 = 16
rD^2 = 225

Taking the square root of both sides of each equation, we find:

rA = 5
rB = 10
rC = 4
rD = 15

Now, let's consider the tangency between the circles. Since the circles are mutually tangent, it means that they all intersect at a single point without overlapping. This implies that the centers of the circles (A, B, C, D) form a rectangle.

The perimeter of a rectangle is given by the formula P = 2(l + w), where P is the perimeter, l is the length, and w is the width.

In our case, since the sides of the rectangle are formed by the radii of the circles, we can determine the length and width as follows:

Length (l) = 2(rA + rB) = 2(5 + 10) = 30 units
Width (w) = 2(rC + rD) = 2(4 + 15) = 38 units

Finally, we can calculate the perimeter of quadrilateral ABCD:

P = 2(l + w) = 2(30 + 38) = 2(68) = 136 units

Therefore, the perimeter of quadrilateral ABCD is 136 units.