Question

Three concentric circles have radii 5,10,15. Chords of the 2 larger circles are tangent to the 2 smaller circles, respectively, and the points of tangency lie on the same radius of the largest circle. Find the area of the trapezoidal region determined by the endpoints of the circle.

Answer 1

We want to find area of trapezoid ABCD.

We are given:
OR = 5
OQ = 10
OP = 15

Therefore we know the altitude (QR) of the trapezoid = OQ - OR = 10 - 5 = 5
We just need to find the two lengths AB and CD.

In triangle OCR, OC=10, OR=5, so use Pythagorus to work out the length of CR. Then CD=2*CR.

In triangle OBQ, OB=15, OQ=10, so use Pythagorus to work out the length of BQ. Then AB=2*BQ.

You can then work out the area of trapezoid ABCD as:\[Area=QR*\frac{AB+CD}{2}\]