Mathematics

Geometry and Trigonometry

115 Questions

Geometry and trigonometry questions cover circles, tangents, and trigonometric ratios. Problems often combine algebraic geometry with angle properties to test spatial reasoning. This forms a core component of the mathematics section in engineering and civil services exams.

Circle tangentsTrigonometric ratiosHyperbola propertiesEllipse tangentsSecant construction

Geometry and Trigonometry Questions

Multiple choice general knowledge

find the ODD one.

  1. diameter

  2. diagonal

  3. tangent

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

Option B is the odd one out. Diameter, tangent, and arc are all terms related to circles, while diagonal is a term related to polygons and other shapes connecting non-adjacent vertices. A circle does not have a diagonal.

Multiple choice general knowledge math & puzzles

A line segment that is tangent to a circle O at its midpoint has its endpoints at A and B. If AB = the diameter of circle O, then triangle OAB is a right triangle.

  1. True

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

The radius drawn to a point of tangency is perpendicular to the tangent line. Since the tangent is perpendicular to radius at the midpoint, and AB is a line segment, angle OAB is 90°, making triangle OAB a right triangle.

Multiple choice

On a map drawn on Gnomonic projection, the great circle course between two points is shown by

  1. a sine curve

  2. a straight line

  3. a parabola

  4. an arc of a circle
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Gnomonic projection is unique because all great circles appear as straight lines on this projection. This property makes it useful for navigation and plotting the shortest distance routes between two points on Earth's surface, since great circles represent the shortest path on a sphere.

Multiple choice maths geometrical construction constructing perpendicular lines perpendicular to a line from an external point constructing an perpendicular line constructing a perpendicular bisector construction of a perpendicular bisector construction of penpendicual bisector set squares

When two lines are perpendicular to each other, the angle is said to be _______ angle.

  1. acute

  2. right

  3. obtuse

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

Two given lines are perpendicular means the angle between them is $90^o$, i.e. a right angle.

Multiple choice maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

Tangents $PA$ and $PB$ drawn to ${ x }^{ 2 }+{ y }^{ 2 }=9$ from any arbitrary point $'P'$ on the line ${ x }+{ y }=25$. Locus of midpoint of chord $AB$ is

  1. $25({ x }^{ 2 }+{ y }^{ 2 })=9(x+y)$
  2. $25({ x }^{ 2 }+{ y }^{ 2 })=3(x+y)$
  3. $5({ x }^{ 2 }+{ y }^{ 2 })=3(x+y)$

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

Let the point on the line $x+y=25$ be $P(a,b)$
Thus equation of chord of contact $AB$ from point $P$ to the circle is given by,
$T  =0 \Rightarrow ax+by = 9$  (i)
Let mid point of $AB$ be $R(h,k)$.
Now equation of chord $AB$ with mid point $R$ is given by,
$T = S _1 \Rightarrow hx+ky = h^2+k^2$ (ii)
Both line (i) and (ii) represents the same line $AB$
$\therefore \displaystyle \frac{a}{h}=\frac{b}{k} = \frac{9}{h^2+k^2}$
$\Rightarrow  a=\cfrac{9h}{h^2+k^2}, b = \cfrac{9k}{h^2+k^2}$
Also point $(a,b)$ lie on the line $x+y = 25$
$\Rightarrow a+b = 25\Rightarrow \cfrac{9h}{h^2+k^2}+ \cfrac{9k}{h^2+k^2}=25$
$ \Rightarrow 25(h^2+k^2) = 9(h+k)$
Hence required locus of $R(h,k)$ is given by $25(x^2+y^2) = 9(x+y)$

Multiple choice maths geometric constructions circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

Given are the steps are construction of a pair of tangents to a circle of radius $4$cm from a point on the concentric circle of radius $6$cm. Find which of the following step is wrong?
(P) Take a point O on the plane paper and draw a circle of radius OA$=4$cm. Also, draw a concentric circle of radius OB$=6$cm.
(Q) Find the mid-point A of OB and draw a circle of radius BA$=$AO. Suppose this circle intersects the circle of radius $4$cm at P and Q.
(R) Join BP and BQ to get the desired tangents from a point B on the circle of radius $6$ cm.

  1. Only (P)

  2. Only (Q)

  3. Both (P) & (Q)

  4. Both (Q) & (R)
Reveal answer Fill a bubble to check yourself
B Correct answer
Multiple choice maths geometric constructions circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

What are the tools required for constructing a tangent to a circle?

  1. ruler

  2. compass

  3. pencil

  4. all the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The tools required for constructing a tangent to a circle is ruler, compass and pencil.

Multiple choice maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

Tangents PA and PB drawn to $ x^2+y^2=9 $ from any arbitrary point 'P ' on the line $ x+y=25 $. Locus of midpoint of chord AB is

  1. $ 25(x^2+y^2)=9(x+y) $
  2. $ 25(x^2+y^2)=3(x+y) $
  3. $ 5(x^2+y^2)=3(x+y) $

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

Let the point on the line $x+y=25$ be $P(a,b)$
Thus equation of chord of contact AB from point P to the circle is given by,
$T  =0 \Rightarrow ax+by = 9$  (i)
Let mid point of AB be $R(h,k)$.
Now equation of chord AB with mid point R is given by,
$T = S _1 \Rightarrow hx+ky = h^2+k^2$ (ii)
Both line (i) and (ii) represents the same line AB
$\therefore \displaystyle \frac{a}{h}=\frac{b}{k} = \frac{9}{h^2+k^2}$
$\Rightarrow  a=\cfrac{9h}{h^2+k^2}, b = \cfrac{9k}{h^2+k^2}$
Also point $(a,b)$ lie on the line $x+y = 25$
$\Rightarrow a+b = 25 \Rightarrow 25(h^2+k^2) = 9(h+k)$
Hence required locus of $R(h,k)$ is given by, $25(x^2+y^2) = 9(x+y)$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons
A polygon has $n$ sides. If all the sides and all the angles are same then this polygon is called a regular polygon. Let ${A} _{1},{A} _{2},{A} _{3},...{A} _{n}$ be a regular polygon of $n$ sides. Let $R$ be the radius of the circumscribed circle of a regular polygon and $r$ be the radius of the inscribed circle of a regular polygon.
If ${A} _{1}{A} _{2}={A} _{2}{A} _{3}={A} _{3}{A} _{4}=...={A} _{n}{A} _{1}=a$

Based on the above information, answer the question:

The area of a regular polygon of $n$ sides is

  1. $\dfrac{n{R}^{2}}{2}\sin{\left(\dfrac{2\pi}{n}\right)}$
  2. $n{R}^{2}\tan{\left(\dfrac{\pi}{n}\right)}$
  3. $\dfrac{n{r}^{2}}{2}\sin{\left(\dfrac{2\pi}{n}\right)}$
  4. $n{r}^{2}\tan{\left(\dfrac{\pi}{n}\right)}$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

The area of a regular polygon of n sides is (where r is inradius, R is circumradius, and a is side of the triangle)

  1. $\displaystyle \frac{nR^{2}}{2}\sin \left ( \frac{2\pi }{n} \right )$
  2. $\displaystyle nr^{2}\tan \left( \frac{\pi }{n} \right )$
  3. $\displaystyle \frac{na^{2}}{4}\cot \frac{\pi }{n} $
  4. $\displaystyle nR^{2}\tan(\frac {\pi}{n})$
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

Area of the regular polygon will be 
$=\dfrac{nR^{2}}{2}sin(\dfrac{2\pi}{n})$.
Now 
$R=\dfrac{s}{2sin(\dfrac{\pi}{n})}$
Hence
$A=\dfrac{ns^{2}}{8sin^{2}\dfrac{\pi}{n}}.2sin(\dfrac{\pi}{n}).cos(\dfrac{\pi}{n})$

$=\dfrac{ns^{2}}{4}.cot(\dfrac{\pi}{n})$. where s is the side of the polygon.

$=nr^{2}.tan(\dfrac{\pi}{n})$ where r is the incentre.

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

  1. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  2. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  3. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $a$ be the radius of the circle 


Then,$\displaystyle s _{1}= $ Area of regular polygon of n sides inscribed in the circle $\displaystyle =\frac{1}{2}na^{2}\sin\left ( \frac{2\pi }{n} \right )$

$\displaystyle s _{2}= $  Area of regular polygon of n sides circumscribing in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$

$\displaystyle s _{3}= $ Area of regular polygon of 2n sides inscribed in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$ 

[replacing $n$ by $2n$ is $\displaystyle {(S _{1}}$]

$\displaystyle \therefore $ Geometric mean of $\displaystyle {S _{1}}$ and 

$\displaystyle {S _{2}}$ $\displaystyle = \sqrt{(S _{1}S _{2})}= na^{2}\sin\left ( \frac{\pi }{n}\right ) = S _{3}$