Question 1189795: A circle with radius 3 cm is tangent to sides PQ, PS, and SR of rectangle PQRS, and passes through the midpoint of diagonal PR. The area of the rectangle PQRS, in cm, is
Answer by math_tutor2020(3817)   (Show Source):
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Drawing
Additional points are- A = midpoint of diagonal PR, and on the circle
- B = midpoint of side PS, and on the circle
- C = center of the circle
- D = point directly above point C, on the circle and side PQ
- E = point directly below point C, on the circle and side SR
Points B, D and E are points of tangency.
The somewhat inscribed circle has radius 3 cm.
The diameter is 2*3 = 6 cm, which is the distance from D to E. Therefore, PS = 6 cm as well.
Furthermore, AB = 6 because all diameters of a circle are the same length.
Point A is halfway between P and Q in terms of horizontal distance. Which indicates that PQ = 2*6 = 12 cm.
We found that
PS = 6
PQ = 12
Therefore the area of the rectangle is
area = length*width
area = PS*PQ
area = 6*12
area = 72
Answer: 72 square cm
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