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

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Last updated date: 20th Jun 2024
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Answer
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Hint: To solve this question, firstly we will make the diagram of the figure according to data given in question. After that, we will find out the area of the rectangle. Then, using the formula of area of quadrant of circle, we will find the area of two quadrants. Then subtracting the area of two quadrants from the area of the rectangle, we will have an area of shaded portion. Then we will find the ratio of rectangle PQRS to the area of the shaded region.

Complete step by step answer:
Let us draw the diagram.
In question we are given that the length of the rectangle is twice the breadth of the rectangle.
So, let the breadth of rectangle be x units.
Then, the length of the rectangle will be equal to 2x units.
Let, L be mid - point on side PQ and P and Q be centre of quadrants and bounded area SLR denotes shaded part.

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Now, we know that area of rectangle is equal to $l\times b$ , 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 = $2x\times x$
On simplification, we get
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 $\pi {{r}^{2}}$, where r is equals to radius of circle, then
Area of the quadrant of the circle will be equal to $\dfrac{1}{4}\pi {{r}^{2}}$.
Now, as L is mid - point of length PQ and PQ = 2x units also, we now that mid – point divides line segments into two equal parts so,
PL = LQ = x units.
So, the radius of the quadrant with centre P and Q will be equal to x units.
So, area of quadrant with centre P = area of quadrant with centre Q = $\dfrac{1}{4}\pi {{x}^{2}}$ units
The, area of both quadrants together will be equals to $\dfrac{1}{4}\pi {{x}^{2}}+\dfrac{1}{4}\pi {{x}^{2}}$
Or, area of both quadrants together $=\dfrac{1}{2}\pi {{x}^{2}}$
Now, the area of the shaded part will be equal to the difference between area of Rectangle and area of summation of two quadrants.
So, Area of shaded part = $2{{x}^{2}}-\dfrac{1}{2}\pi {{x}^{2}}$
On simplification, we get
Area of shaded part = $\dfrac{{{x}^{2}}(4-\pi )}{2}$units
So, the ratio of rectangle PQRS to the area of shaded region = $\dfrac{2{{x}^{2}}}{\dfrac{{{x}^{2}}(4-\pi )}{2}}$
On simplification, we get
$\Rightarrow$ Ratio = $\dfrac{2}{\dfrac{(4-\pi )}{2}}$
$\Rightarrow$ Ratio =$\dfrac{4}{(4-\pi )}$
$\Rightarrow$ Ratio =$\dfrac{4}{\left( 4-\dfrac{22}{7} \right)}$
$\Rightarrow$ Ratio =$\dfrac{4}{\left( \dfrac{28-22}{7} \right)}$
$\Rightarrow$ Ratio =$\dfrac{28}{6}$
$\Rightarrow$ Ratio = $\dfrac{14}{3}$

Hence, the ratio of rectangle PQRS to the area of shaded region is equals to 14 : 3.

Note: To solve such questions one must know the formulas such as area of rectangle is equals to $l\times b$ , where l denotes length of rectangle and b denotes breadth of rectangle, area of circle is equals to $\pi {{r}^{2}}$, where r is equals to radius of circle and area of quadrant of circle will be equals to $\dfrac{1}{4}\pi {{r}^{2}}$. Always remember that the ratio has no units and if we have a fraction $\dfrac{a}{b}$, then ratio is a : b. Try not to make any calculation errors.