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Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors
$\triangle ABC$ 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. $120^\circ$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By inscribed angle theorem, 

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

Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

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. $\displaystyle \frac{\pi }{1524}$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $\pi / 1016$

Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

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
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circumference of a circle corresponds to an angle of 360 degrees at the center. Since the minor arc length is 1/4 of the total circumference, the central angle subtended by this arc is (1/4) * 360 degrees = 90 degrees.

Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

Let a semicircle with centre O and diameter AB. Let P and Q be points on the semicircle and R be a point on AB extended such that OA =QR < PR. If $\widehat{POA} = 102^0$ then $\widehat{PRA} $ is 

  1. $51^0$
  2. $34^0$
  3. $25.5^0$

  4. not possible to be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation


Since R can be any point between A & R' & hence its corresponding point Q will lie on the arc AQ'. Hence, $\angle$ PRA can not be uniquely determined.