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Angle subtended by arc - class-VIII

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State true or false.
Sector is the region between the chord and its corresponding arc.

  1. True

  2. False


Correct Option: B
Explanation:

Sector is the region between an arc and two radii joining. The centre to the end points of the arc.

False.

$r$ is the radius and $l$  is the length of an arc. The area of a sector is ______.

  1. $\dfrac { 1 } { 2 } r l$

  2. $\dfrac { 3 } { 2 } r ^ { 2 } l$

  3. $\dfrac { 4 } { 3 } r l$

  4. $\dfrac { 3 } { 2 } r l$


Correct Option: A
Explanation:

$\begin{array}{l} Area\, of\, a\, \sec  tor\, =\dfrac { 1 }{ 2 } { r^{ 2 } }\theta  \ =\dfrac { 1 }{ 2 } \times r\times r\theta  \ =\dfrac { 1 }{ 2 } \times r\times l \ Hence,\, option\, A\, is\, \, the\, \, correct\, \, answer. \end{array}$

State true or false:


Sector is the region between the chord and its corresponding arc.

  1. False

  2. True 

  3. cannot be determined

  4. none of the above


Correct Option: A
Explanation:

Segment is the region between the chord and its corresponding arc.

Say true or false:

A sector is a region between the chord and its corresponding arc.

  1. True

  2. False


Correct Option: B
Explanation:

False.
The region between an arc and the two radii, joining the centre to the end points of the arc is called a sector

A circular disc of radius $10 cm$ is divided into sectors with angles $120^o$ and $150^o$, then the ratio of the area of two sectors is

  1. $4 : 5$

  2. $5 : 4$

  3. $2 : 1$

  4. $8 : 7$


Correct Option: A
Explanation:

Area of sector formed from angle $\theta=\dfrac{\theta}{360^\circ}\pi r^2$, where $r$ is the radius of the circle
Now, if angle is $120^\circ$, $150^\circ$ then the ratio of area of sector will be

$\Rightarrow\dfrac{\dfrac{120^\circ}{360^\circ}\pi r^2}{\dfrac{150^\circ}{360^\circ}\pi r^2}$

$\Rightarrow \dfrac{4}{5}$
Hence, the required ratio is $4:5$. 

The region between an arc and two radii joining the centre to the end points of the arc is called

  1. sector

  2. segment

  3. semicircle

  4. non of these


Correct Option: A
Explanation:

The region between an arc and two radii joining the center to the end points of the arc is called sector.The minor are corresponds to minor sector  and  major arc correspond to major sector.

$\triangle ABC$ is inscribed in a circle. Point $P$ lies between $A$ and $C$, whereas point $Q$ lies between $B$ and $C$. If $m(\text{arc}\, APC) = 60^\circ$ and $\angle BAC = 80^\circ$, find $m(\text{arc}\, BQC)$.
  1. $180^\circ$

  2. $90^\circ$

  3. $160^\circ$

  4. $120^\circ$


Correct Option: C
Explanation:

By inscribed angle theorem, 

$ \cfrac 12 m\angle BAC = m(arc BQC)$
$m(arc BQC) = 2 \times \angle BAC$
$\therefore m(arc BQC) = 2 \times 80^o = 160^o$

Which of the following is not a sector of a circle?

  1. Pizza slice

  2. Cathedral window

  3. Birthday cap

  4. Apple pie


Correct Option: C
Explanation:

Pizza slice, Cathedral window and apple pie are sector of a circle 

whereas Birthday cap is of conical shape which is not a sector of circle.
Hence option 'C' is correct choice 

Circular dome is a 3D example of which kind of sector of the circle?

  1. Quadrant

  2. Semicircle

  3. Octant

  4. Sextant


Correct Option: B
Explanation:

A circular dome is a 3 dimension structure of a semicircle.

Points $A,B,C $ are on a circle, such that $m(arc AB)=m(arc BC)=^o$. No point, except point $B$, is common to the arcs.which is the type of $\triangle ABC$?

  1. Equilateral triangle

  2. Scalene triangle

  3. Right angled triangle

  4. Isosceles triangle


Correct Option: A

Find the area of a sector of a circle with radius $6$cm if angle of the sector is $60^o$.

  1. $6\pi$

  2. $7\pi$

  3. $3\pi$

  4. $5\pi$


Correct Option: A
Explanation:
Area of sector of circle $=\dfrac{\theta}{360}\times \pi r^2$
$=\dfrac{60}{360}\times \pi\times (6)^2$
$=\dfrac{1}{6}\times \pi\times (36)$
$=6\pi$.

Consider a circle with unit radius. There are seven adjacent sectors, $S _1, S _2, S _3, ............ S _7$, in the circle such that their total area is $\dfrac {1}{8}$ of the area of the circle. Further, the area of the $j^{th}$ sector is twice that of the $(j-1)^{th}$ sector, for $j$ $=$ $2, ........... 7$. What is the area of sector $S _1?$

  1. $\displaystyle \frac{\pi }{508}$

  2. $\displaystyle \frac{\pi }{2040}$

  3. $\displaystyle \frac{\pi }{1016}$

  4. $\displaystyle \frac{\pi }{1524}$


Correct Option: C
Explanation:

Let the area of thesector S$ _1$ be x units. Then, the area of the corresponding sectors shall be 2x, 4x, 8x, 16x, 32x and 64x. The total area then shall be 127x units. This is $\displaystyle \frac{1}{8}$ of the total area of the circle. 

Hence, the total area of the circle will be $127x \times 8 = 1,016 x\ units.$
$\Rightarrow 1016 x = \pi (1)^2 \Rightarrow x = \pi/1016$
Hence area of sector $S _1 $ is $\pi / 1016$

The angle subtended by the chord AB in the minor arc of S is - 

  1. $\dfrac{3 \pi}{4}$

  2. $\dfrac{5 \pi}{6}$

  3. $\dfrac{2 \pi}{3}$

  4. $\dfrac{ \pi}{4}$


Correct Option: A

The length of minor arc $\overset{\frown}{AB}$ of a circle is $\dfrac{1}{4}$ of its circumference, then the measure of the angle subtended by the minor arc $\overset{\frown}{AB}$ will be ....

  1. 30

  2. 45

  3. 90

  4. 60


Correct Option: A

With a given centre and a given radius,only one circle can be drawn.

  1. True

  2. False


Correct Option: A

If angle of sector is $x^o$, then formula used to calculate area is

  1. $\dfrac{x^o}{360}\times \pi r^2$

  2. $2\dfrac{x^o}{360}\times \pi r$

  3. $\dfrac{x^o}{180}\times \pi r^2$

  4. $2\dfrac{x^o}{360}\times r^2$


Correct Option: A
Explanation:

If angle of sector is $x^o$ then formula used to calculate is $\dfrac{x^o}{360}\times \pi r^2$.

If the circumference of a circle is $8$ units and arc length of major sector is $5$ units then find the length of minor sector.

  1. $3$ units

  2. $5$ units

  3. $7$ units

  4. None of these


Correct Option: A
Explanation:

Length of major arc + Length of minor arc = Circumference

Length of Minor arc $= 8 – 5 = 3$ units 

The angle subtended at the centre of a circle of radius $3cm$ by an arc of length $1cm$ is:

  1. $\cfrac { { 30 }^{ o } }{ \pi } $

  2. $\cfrac { { 60 }^{ o } }{ \pi } $

  3. ${ 60 }^{ o }$

  4. None of the above


Correct Option: B
Explanation:

Angle subtended at the centre  of circle is $\theta =\dfrac { l }{ r }$ 

$\Rightarrow \theta =\dfrac { 1 }{ 3 }$  
Now, $\pi$ radian $ =180^{o}$ 
$\Rightarrow \frac { 1 }{ 3 }$ radian $=180^{o}\times \dfrac { 1 }{ 3\pi  } =\dfrac { 60^{o} }{ \pi  }$ 
Hence, option B is correct.

Write True or False:

The tangent to the circumcircle of an isosceles $\triangle ABC$ at A, in which $AB = AC$, is parallel to BC.

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient


Correct Option: A
Explanation:

Given-

PQ is a tangent to a circle at a point A when the circle is a circumcircle of the isosceles $\Delta ABC$.
$ AB=AC.$ 
To find out -
The statement, $PQ\parallel BC$, is true or not.
Justification-
In $\Delta ABC$ we have,
$ AB=AC.$ 
$\therefore  \angle ABC=\angle ACB$    ...(base angles of an isosceles triangle)    ........(i)
Again, PQ is the tangent to the circle at A & AB is a chord drawn from A. 
And AB subtends \angle ACB to the corresponding alternate segment of the circle.
$ \therefore  \angle PAB=$ corresponding alt. segment $\angle ACB$ ......(ii)
So, from (i) & (ii), 
$\angle PAB=\angle ABC.$ 
But they are alternate angles $\Longrightarrow  PQ\parallel BC$ 
$\therefore$ The statement, $PQ\parallel BC$, is true.

In $\bigodot (P, 6)$, the length of an arc is $\pi$. Then the arc subtends an angle of measure ___at the center.

  1. 30

  2. 60

  3. 90

  4. 120


Correct Option: A
Explanation:

Given : $r=6$

$l=\pi $
We know that $l=r\theta $
$\Rightarrow \cfrac { \pi  }{ 6 } =\theta $
$\theta ={ 30 }^{ 0 }$
$\therefore$ The arc subtends ${ 30 }^{ 0 }$ at $P$.

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