Three circles with radii 5, 10, and 15 ft are externally tangent to one another, as shown in the figure. find the area of the sector of the circle of radius 5 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.

  (25/4)π ft²

Step-by-step explanation:

The central angle of the sector of interest is the angle opposite the longest side of the triangle joining the centers of the circles. That triangle has side lengths of 5+10=15, 5+15=20, and 10+15=25. Those lengths are in the ratio ...

  15 : 20 : 25 = 3 : 4 : 5

so the triangle is recognizable as a right triangle. Its largest angle is 90°, so the sector has an area that is 1/4 the area of a circle of radius 5.

  A = (1/4)π(5 ft)^2 = (25/4)π ft^2

The area is 6.25π square feet.