PQRS is a rectangle in which length is two times the breadth and L is the midpoint of side PQ. With P and Q as centre, draw two quadrants. Find the ratio of rectangle PQRS to the area of the shaded region.

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14:3

13:4

12:3

12:5

answer is A.

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Detailed Solution

Let's draw a diagram.
The point is that the length of the rectangle is twice the width of the rectangle.
So let the width of the rectangle be x units.
Then the length of the rectangle will be equal to 2x units.
Let L be the midpoint on the side PQ and P and Q the midpoints of the quadrants and the bounded region SLR denote the shaded portion.

Now, we know that area of rectangle is equal to

l × b

,where l denotes length of rectangle and b denotes breadth of rectangle.
For rectangle PQRS, we have breadth b = x and length l = 2x
So, area of rectangle PQRS =

2 x × x

Area of rectangle PQRS =

2 x 2

Now, we know that, quadrant is one-fourth part of a circle.
So, if area of circle is equals to

π r 2

, where r is equals to radius of circle, then,
The area of the quadrant of the circle will be equal to

.
Now, since L is the midpoint of the length of PQ and PQ = 2x units, now this midpoint divides the line segments into two equal parts, so,
PL = LQ = x units.
So the radius of the quadrant centered at P and Q will be equal to x units.
So, area of quadrant with centre P = area of quadrant with centre Q =