Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

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 maths similarity relation between perimeters of similar shapes basic proportionality theorem and its converse basic proportionality theorem

The areas of two similar triangle are $18\ cm^{2}$ and $32\ cm^{2}$ respectively. What is the ratio of their corresponding sides?

  1. $3:4$
  2. $4:3$
  3. $9:16$
  4. $16:9$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The ratio of areas of similar triangles is the square of the ratio of their corresponding sides. sqrt(18/32) = sqrt(9/16) = 3/4.

Multiple choice maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

The ratio of the lengths of the corresponding sides of $2$ similar right angled triangles is $2:5$. If the length of the hypotenuse of the smaller triangle is $5$ inches, find the length of the hypotenuse of the larger triangle (in inches):

  1. 2

  2. 2.5

  3. 7

  4. 10

  5. 12.5
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Ratio of the length of the sides of the two triangle $=2:5$

If hypotenuse  of small triangle $=5$ inches
Let the hypotenuse of  larger triangle $=x$
$\therefore \dfrac{5}{x}=\dfrac{2}{5}$
$\therefore  x=\dfrac{25}{2}=12.5$  inches

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 mathematical modelling proof by contradiction similar triangles

To prove: "The perpendicular from centre of a circle to the chord, bisects the chord." The proof started from assumption "Let OM be the perpendicular to chord AB". 

This method of proof is  

  1. The proof by contradiction

  2. The Direct method.

  3. Induction method.

  4. The proof by contrapositive method.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In direct method we have to assume the hypothesis as we have assumed the hypothesis "Let OM be the perpendicular to chord AB" in this case hence the option is $B$

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

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

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

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

A sector of a circle with sectorial angle of $\displaystyle 36^{\circ} $ has an area of 15.4 sq cm The length of the arc of the sector is

  1. $8.8 m$
  2. $4.4 m$
  3. $0.22 m$
  4. $0.44 m$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \frac{36}{360}\times \frac{22}{7}r^{2}=15.4\Rightarrow r^{2} =\frac{15.4\times 5\times 7}{11}=49$
$\displaystyle \Rightarrow r=7$
$\displaystyle C=2\pi r=2\times \frac{22}{7}\times 7=44cm$
$\displaystyle =0.44 m$

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

In a circle of radius 21 cm an arc subtends an angle of $\displaystyle 56^{\circ} $ at the centre of the circle. The length of the arc is

  1. $20.53$ cm
  2. $17.53$ cm
  3. $15.53$ cm
  4. $16.53$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle \theta =56^{\circ},r=21cm$
Length. of $\displaystyle AB=\dfrac{56^{\circ}}{360^{\circ}}\times 2\times \dfrac{22}{7}\times 21$
$\displaystyle =\dfrac{616}{30}=20.53cm$

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

If an arc of a circle of radius 14 cm subtends an angle of $60^{\circ}$ at the centre, then the length of the arc is $\displaystyle \frac{44}{3} cm$.

  1. True

  2. False

  3. Niether

  4. Either
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Arc length = (theta / 360) * 2 * pi * r. Here, (60 / 360) * 2 * (22 / 7) * 14 = (1 / 6) * 2 * 22 * 2 = 88 / 6 = 44 / 3 cm. The statement is true.

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

The length of an arc of a sector of a circle of radius r units and of centre angle $\theta$ is $\displaystyle \frac{\theta}{360^{\circ}} \times \pi r^2$.

  1. True

  2. False

  3. Neither

  4. Either
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The formula provided, (theta / 360) * pi * r^2, is the formula for the area of a sector, not the length of an arc. The arc length formula is (theta / 360) * 2 * pi * r.

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

Length of an arc of a circle with radius $r$ and central angle $\theta$ is(angle in radians):

  1. $\dfrac{r\times \theta}{360^{o}}$
  2. $\dfrac{r\times \theta}{180^{o}}$
  3. $\dfrac{r\times \theta}{90^{o}}$
  4. $r\times \theta$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let $r$ be the radius of a circle and $\theta$ be the central angle
Length of an arc of the sector $=r\times \theta$
Hence, length of an arc of a circle $=r\times \theta$.