Area Related to Circles Test

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Area Related to Circles Test - 30

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Question 1
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Five identical rectangles are placed inside a square with side $$24 cm$$ as shown in the diagram. What is the area of one rectangle ?

A
$$12 \mathrm { cm } ^ { 2 }$$

B
$$16 \mathrm { cm } ^ { 2 }$$

C
$$24 \mathrm { cm } ^ { 2 }$$

D
$$32 \mathrm { cm } ^ { 2 }$$

Solution

Consider vertical line, let $$b$$ & $$h$$ be breadth and height of rectangle then,

$$24=b+h+h+b$$ (vertically)
$$b+h=12 \ \ \ \ \ \ ....(1)$$

$$24=h+(h-b)+b+h$$ (horizontally)
$$3h=24$$

$$h=8$$

Substituting $$h=8$$ in equation $$(1) $$

$$b=4$$

Area $$= bh=32 \ cm^2$$.
Question 2
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In the following figure $$A B = B C$$ and $$A C = 84 \mathrm { cm } .$$ The radius of the smallest circle is $$14\mathrm { cm } .$$ $$B$$ is the centre of the largest semicircle. What is the area of the shaded region.

A
$$335 \mathrm { cm } ^ { 2 }$$

B
$$770 \mathrm { cm } ^ { 2 }$$

C
$$840 \mathrm { cm } ^ { 2 }$$

D
$$650 \mathrm { cm } ^ { 2 }$$

Solution

REF.Image
AC = 84 cm
AB = BC =
2AB = 84
AB = 42
Area of circle (semi) whose diameter is AB.
$$A_{1}=\pi R^{2}/2$$
$$=\pi (21)^{2}/2=\frac{22}{7}\times 22\times 21\times \frac{1}{2}$$
$$= 11\times 3\times 21=33\times 21$$
$$= 693 cm^{2}$$
Area of circle (semi) whose diameter is BC
$$A_{2}=\frac{\pi R^{2}}{2}$$
$$= 693 cm^{2}$$
Area of circle whose diameter is AC
$$A=\frac{\pi R^{2}}{2}=\frac{22}{7}\times 42\times 42\times \frac{1}{2}$$
$$A=\frac{5544}{2}cm^{2}= 2772 cm^{2}$$
Area of smallest circle
$$a=\pi r^{2}=\frac{22}{7}\times 14\times 14=616 cm^{2}$$
Area of the shaded region = $$A-(A_{1}+A_{2}+a)$$
$$=2772-(693+693+616)$$
$$=2772-2002$$
$$770 cm^{2}$$
Question 3
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The area $$(in cm^2)$$ of a sector of a circle with an angle of $$45^o$$ and radius $$3$$ cm is_.

A
$$3\dfrac{13}{14}$$

B
$$4\dfrac{13}{14}$$

C
$$3\dfrac{51}{56}$$

D
$$3\dfrac{15}{28}$$

Solution

Given circle radius $$=3cm$$ (r)
Sector angle $$=45$$
Area of sector $$=\dfrac{45}{360}\times \pi r^2$$
$$=\dfrac{1}{8}\times \dfrac{22}{7}\times (3)^2$$
$$=\dfrac{22}{8.7}\times 9$$
$$=\dfrac{99}{28}$$ sq. cm
$$=3\dfrac{15}{28}$$ sq. cm.
Question 4
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In a circle of radius $$\displaystyle 14 $$ cm an arc subtends an angle of $$\displaystyle 36^0$$ at the centre. The length of the arc is

A
$$\displaystyle 6.6 $$ cm

B
$$\displaystyle 7.7 $$ cm

C
$$\displaystyle 8.8 $$ cm

D
$$\displaystyle 9.1 $$ cm

Solution

$$\displaystyle \theta = \left( 36 \times \frac{\pi}{180}\right)^c = \left( \frac{\pi}{5} \right)^c$$ and $$\displaystyle r = 14 $$ cm.
$$\displaystyle \therefore \ \ l = r \theta = \left( 14 \times \frac{\pi}{5} \right)$$ cm = $$\displaystyle \left( 14 \times \frac{22}{7} \times \frac{1}{5} \right)$$ cm = $$\displaystyle \frac{44}{5} $$ cm = $$\displaystyle 8.8 $$ cm.
Question 5
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The center of the circle lies

A
in the interior of the circle

B
in the exterior of the circle

C
on the circle

D
None of these

Solution
Question 6
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The shaded portion of the circle represents

A
Semicircle

B
Diameter

C
A sector of the circle

D
Segment
Question 7
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In the given figure, $$r$$ represents

A
Radius

B
Diameter

C
Tangent

D
Chord

Solution

$${\textbf{Step -1: Find r.}}$$

$${\text{The distance from the center point to any point on the circle is called the radius of a circle}}$$

$${\text{Here, r is the radius in the figure.}}$$

$${\textbf{Hence, the correct answer is option A.}}$$
Question 8
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In the adjoining figure, $$18 in$$ is the length of

A
Radius

B
Diameter

C
Any chord other than the diameter

D
None of these

Solution
Question 9
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A chord divides a circle into two

A
Diameter

B
Semicircle

C
Sectors

D
Segments

Solution
Question 10
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The centres of a set of circle,

each of radius $$ 3$$, lies on the circle $$x^2 + y^2 = 25$$. The locus of any

point in the set is

A
$$4\leq x^2+y^2\leq 64$$

B
$$ x^2+y^2\leq 25$$

C
$$ x^2+y^2\geq 25$$

D
$$3\leq x^2+y^2\leq 9$$