Mathematics · Quantitative Aptitude

Circle and Arc Properties

115 Questions

Circle and arc properties involve calculating arc lengths, understanding radius relationships, and solving geometric proofs. These geometry concepts are crucial for quantitative aptitude tests. Review these questions to improve your spatial reasoning and accuracy.

Arc length calculationsCircle theoremsRadius and diameterInscribed polygonsCentral angles

Circle and Arc Properties Questions

Multiple choice physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

The actual distance moved along the circle will be                   the distance moved by the projection on the diameter.

  1. less than

  2. equal to

  3. greater than

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius of the circle be $R$.

Thus distance moved along the circle  $D = \pi R = 3.14 R$
Projection of this distance along the diameter is equal to the diameter of the circle i.e.  $2R$
Thus actual distance moved along the circle will be greater than the distance moved by the projection on the diameter.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

When constructing an inscribed regular hexagon, how will you choose the arc measurement?

  1. radius of the circle

  2. diameter of the circle

  3. chord of the circle

  4. circumference of the circle
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

When constructing an inscribed regular hexagon in a circle, we choose radius of the circle as a arc measurement.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

How many equal parts you will cut the circle to draw inscribing hexagon?

  1. $4$
  2. $5$
  3. $6$
  4. $7$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Hexagon is a $6$-sided polygon.
So we will cut the circle into $6$ equal parts.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

The centre of the circle circumscribing the square whose three sides are $3x+y=22,x-3y=14$ and $3x=y=62$ is:

  1. $\left( \dfrac { 3 }{ 2 } ,\dfrac { 27 }{ 2 } \right) $
  2. $\left( \dfrac { 27 }{ 2 } ,\dfrac { 3 }{ 2 } \right) $
  3. $(27,3)$
  4. $\left( 1,\dfrac { 2 }{ 3 } \right) $
Reveal answer Fill a bubble to check yourself
B Correct answer
Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

In regular hexagon, if the radius of circle through vertices is r, then length of the side will be

  1. $\displaystyle \frac{2\pi r}{6}$

  2. r
  3. $\displaystyle \frac{\pi r}{6}$
  4. $\displaystyle \frac{r}{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$   Radius of a circle is $r$.

$\Rightarrow$   In regular hexagon all sides are equal.
$\Rightarrow$   The regular hexagon has 6 equilateral triangles. The diameter of the circle is $2r$ in this case, will coincide with 2 equilateral triangles. So the side of the hexagon will be $r$.
$\therefore$   Length of side of hexagon is $r$.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

When constructing the circles circumscribing and inscribing a regular hexagon with radius $3$ m, then inscribing hexagon length of each side is

  1. $1m$
  2. $2m$
  3. $3m$
  4. $4m$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When constructing the circles circumscribing and inscribing a regular hexagon with radius $3$ m, then inscribing hexagon length of each side is $3$ m.

Multiple choice maths fundamental concepts - geometry terms related to polygons curves open and closed figures

The straight line AB is divided at C so that $\bar{AC} = 3\bar{CB}$. Circles are described on AC and CB as diameters and a common tangent meets AB produced at D. Then $\bar{BD}$ equals.

  1. the diameter of the smaller circle

  2. the radius of the smaller circle

  3. the radius of the larger circle
  4. $\bar{CB} \sqrt{3}$

  5. the difference of the two radii
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $ x=\overline{BD} $ and let $ r$ be the radius of the small circle. 


Draw the line from the center of each of the circles to the point of contact of the tangent of the circle. 

By similar triangles, 

$ \dfrac{x+r}{r}=\dfrac{x+5r}{3r} \implies x=r$.

$ \overline{BD} $ equals the radius of the smaller circle.

Multiple choice mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

The angle subtended at the centre of circle of radius $3$ metres by an arc of length $1$ metre is equal to

  1. $20^\circ $
  2. $60^\circ $
  3. $\dfrac{1}{3}\,radian$
  4. $\,3\,radian$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that 

$l=r\times\theta$

Where $l\rightarrow arc$ $length$
            $r\rightarrow radius$
            $\theta\rightarrow angle$ $subtended$ $by$ $the$ $arc$

Substituting the values of these terms we get,

$\Rightarrow 1=3\times\theta$

$\Rightarrow\theta=\dfrac{1}{3} radian$

Multiple choice mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

The value of $\displaystyle 144^{\circ}$ in circular measure is ___ 

  1. $\displaystyle \frac{3\pi ^{c}}{4}$
  2. $\displaystyle \frac{2\pi ^{c}}{3}$
  3. $\displaystyle \frac{4\pi ^{c}}{5}$
  4. $\displaystyle \frac{5\pi ^{c}}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ {144}^{0} = {144}^{0} \times \dfrac {{\pi}^{c}}{{180}^{0}} = \dfrac {4{\pi}^{c}}{5} $

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The greatest possible number of points of intersection of 8 straight lines and $4$ circles is $104$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to question,

There are 8 lines, for two lines meet in a point,

=>  $^8C _2\times 1= \dfrac{8.7}{1.2}=28$

Line and circle meet in two points,

=>$(^8C _1\times ^4C _1) \times 2 =64$

Two circles meet in two points,

=>  $(^4C _2)\times2 =\dfrac{4.3}{1.2}.2= 104$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

There are 6 equally spaced points A, B, C, D, E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points advertises?

  1. $^6P _5$
  2. $1$
  3. $55$
  4. $42$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For any 5 points chosen from 6 equally spaced points on a circle, the area of the resulting convex pentagon is determined by the relative positions of the points. Due to the symmetry of the points on the circle, all convex pentagons formed by choosing 5 out of 6 points are congruent and thus have the same area.

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

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
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

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