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In the diagram above, the sides of rectangle ABCD have a ratio AB : BC = 1 : 2, and the circle is tangent to three sides of the rectangle. If a point is chosen at random inside the rectangle, what is the probability that it is not inside the circle?
  • a)
  • b)
  • c)
  • d)
  • e)
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
In the diagram above, the sides of rectangle ABCD have a ratio AB : BC...
We aren’t given any absolute lengths.  For convenience, I am going to assume that the radius of the circle is r = 1.  That’s very easy.  Right away, we know the area of the circle is <m>pi</m>.  Notice, the height of the triangle is equal to the diameter of the circle, so h = AB = 2.  We are told the ratio of AB : BC = 1 : 2, so BC, the width, must equal w = BC = 4.  Area of the rectangle is h*w = (AB)*(BC) = 8.  That, right there, is our “denominator area”.   Now, for the numerator area, the area of the rectangle that does not include the circle, subtract the circle from the rectangle: A = <m>8 – pi</m>.  That’s our numerator.
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In the diagram above, the sides of rectangle ABCD have a ratio AB : BC = 1 : 2, and the circle is tangent to three sides of the rectangle. If a point is chosen at random inside the rectangle, what is the probability that it isnotinside the circle?a)b)c)d)e)Correct answer is option 'D'. Can you explain this answer?
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