Mathematics · Quantitative Aptitude

Coordinate Geometry and Construction

139 Questions

Coordinate geometry involves finding midpoints, calculating ratios of line segments, and understanding 3D projections. These concepts are crucial for tackling quantitative aptitude sections. Practice these questions to master geometric constructions and coordinate plotting.

Midpoint calculationsSection formulas3D geometry projectionsGeometric construction stepsLine segment ratios

Coordinate Geometry and Construction Questions

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

To construct a perpendicular to a line($L$) from a point ($P$) outside the line, steps are given in jumbled form.Identify the fourth step from the following
1) Draw line $PQ$
2)Draw a line $L$ and consider point $P$ outside the line
3)Take P as a center, draw $2$ arcs on line $L$ and name it as points $A$ and $B$ respectively
4)Taking $A$ and $B$ as a center one by one and keeping the same distance in compass, draw the arcs on other side of the line.The point where these arcs intersect name that point as $Q$

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

The correct sequence is:

Step 1. Draw a line $L$ and consider a point $P$ outside the line.
Step 2. Take $P$ as center and draw two arcs on line $L$ ans name the points $A$ and $B$ respectively.
Step 3.Taking $A$ and $B$ as centres one by one and keeping the same distance in compass , draw the arcs on other side of the line .The point where these arcs intersect name that as $Q$
Step 4. Draw line $PQ$
So the fourth step is $1$
Option $D$ is correct.

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The ordinate of the point which divides the lines joining the origin and the point $(1,2)  $ externally in the ratio of $3:2$ is

  1. $-2$
  2. $ \displaystyle \frac{3}{5} $
  3. $ \displaystyle \frac{2}{5} $
  4. $6$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Co-ordinates of the required point will be
$\displaystyle y=\frac{m _{1}y _{2}-m _{2}y _{1}}{m _{1}-m _{2}}=\frac{3\times 2-2\times 0}{3-2}=6$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The points (22,23) divides the join of P (7,5) and Q externally in the ratio 3:5, then Q=

  1. $(3,7)$
  2. $(-3,7)$
  3. $(3,-7)$
  4. $(-3,-7)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$\left(22,23\right)$dives $P\left(7,5\right)$ and $Q$ in the ratio $3:5$ externally.
Let $Q\left(x,y\right)$
$22=\dfrac{3x-5\times 7}{-2}$
$-44=3x-35$
$x=-3$
$23=\dfrac{3y-5\times 5}{-2}$
$y=-7$
$Q\left(-3,-7\right)$
$D$ is correct.
Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The point $(22, 33)$ divides the join of $P(7, 5)$ and $Q$ externally in the ratio $3 : 5$, then coordinates of $Q$ are

  1. $(3, 7)$
  2. $( - 3, -7)$
  3. $(3, - 7)$
  4. $( - 3, \frac{ - 41}{3})$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let the point $Q$ be $\left(x,y \right)$
As the point $\left(22,23 \right)$ diverts the line joining point $P\left(7,5 \right)$ and $Q \left(x,y \right)$ in $3:5$ externally.
So.  
$22=\dfrac{ 3\left(x\right)-5\left(7\right)} {3-5}\quad 23=\dfrac{ 3\left(y\right)-5\left(5\right)} {3-5}$
$22=\dfrac{3x-35}{-2}\quad 23=\dfrac{3y-25}{-2}$
$-44=3x-35\quad -46=3y-25$
$-=3x\quad -21=3y$
$x=-3\quad y=\dfrac{-21}{3}=-7$
Point $Q=\left(-3,-7\right)$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

Find the co-ordinates of the point $P$ which divides segment $JL$ externally in the ratio $m:n$ in the following example:

$J(5, -3), L(0, 9), m:n = 4:3$

  1. $(-15, 45)$
  2. $(15, -45)$
  3. $(15, 45)$
  4. $(-15, -45)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let the co-ordinates of the point $P$ be $P\left( x,y \right)$  which divides the line segment $JL$ joining the points $J\left( { x } _{ 1 },{ y } _{ 1 } \right) =\left( 5,-3 \right)  \&  L\left( { x } _{ 2 },{ y } _{ 2 } \right) =\left( 0,9 \right)$ in the ratio $m:n=4:3$

Then, by the section formula, $ x=\dfrac { n{ x } _{ 1 }-m{ x } _{ 2 } }{ n-m } =\dfrac { 3\times 5-4\times 0 }{ 3-4 } =-15$ and 
$ y=\dfrac { n{ y } _{ 1 }-m{ y } _{ 2 } }{ n-m } =\dfrac { 3\times (-3)-4\times 9 }{ 3-4 } =45$ 
$\therefore  P\left( x,y \right) =P\left( -15,45 \right)  $
Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The co-ordinates of the point B which divides segment PQ joining the points $P(-2,-4)$ and $Q(-2,-1)$ externally in the ratio $m: n=7:1$ are

  1. $ B(x,y)=\left( 2,-\dfrac { 1 }{ 2 } \right) $
  2. $ B(x,y)=\left( -2,\dfrac { 1 }{ 2 }\right) $
  3. $ B(x,y)=\left( -2,-\dfrac { 1 }{ 2 } \right) $
  4. $ B(x,y)=\left( 2,\dfrac { 1 }{ 2 }\right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given that- the point $B\left( x,y \right)$ externally divides the line segment PQ joining the points $P(-2,-4)$ & $Q\left( { x } _{ 2 },{ y } _{ 2 } \right) =(-2,-1)$ in the ratio $m:n=7:1$

To find out- the coordinates of B.
Solution-  We know that if a point $B\left( x,y \right)$ externally divides the line segment PQ joining the points $P\left( { x } _{ 1 },{ y } _{ 1 } \right)$ & 
$Q\left( { x } _{ 2 },{ y } _{ 2 } \right)$ in the ratio $m:n$ then, by the section formula, 
$x=\dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n }$ &  $y=\dfrac { m{ y } _{ 2 }-n{ y } _{ 1 } }{ m-n }$
Here ${ x } _{ 1 }=-2, { x } _{ 2 }=-2, { y } _{ 1 }=-4, { y } _{ 2 }=-1, m=7, n=1$
$ \therefore  x=\dfrac { 7\times \left( -2 \right) -1\times \left( -2 \right)  }{ 7-1 } =-2$ 
And $y=\dfrac { 7\times (-1)-1\times \left( -4 \right)  }{ 7-1 } =-\dfrac { 1 }{ 2 }$
$\therefore  B(x,y)=\left( -2,-\dfrac { 1 }{ 2 }  \right)$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The co-ordinates of the point B which divides segment PQ joining the points $P(-2,-4)$ and $Q(-2,-1)$ in the ratio $m:n = 2 : 5$,  are

  1. $B(x,y)=(2, 6)$
  2. $B(x,y)=(-2,6)$
  3. $B(x,y)=(2,-6)$
  4. $B(x,y)=(-2,-6)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given that:

Point $B\left( x,y \right)$  externally divides the line segment $PQ$ joining the points $P(-2,-4)$ &  $Q\left( { x } _{ 2 },{ y } _{ 2 } \right) =(-2,-1)$ in the ratio $m:n=2:5$. 

To find out- the co-ordinates of B.

Solution-
We know that if a point $B\left( x,y \right)$ externally divides the line segment PQ joining the points $P\left( { x } _{ 1 },{ y } _{ 1 } \right)$ & $Q\left( { x } _{ 2 },{ y } _{ 2 } \right)$ in the ratio $m:n$ then, by the section formula, 
$x=\dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n }$ & $y=\dfrac {my _2-ny _1}{m-n}$

Here, ${ x } _{ 1 }=-2, { x } _{ 2 }=-2, { y } _{ 1 }=-4, { y } _{ 2 }=-1, m=2, n=5.$

$\therefore  x=\dfrac { 2\times (-2)-5\times (-2) }{ 2-5 } =-2$ and $y=\dfrac { 2\times (-1)-5\times (-4) }{ 2-5 } =-6$

$\therefore  B(x,y)=\left( -2,-6\right)$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

Find the co-ordinates of the point dividing the join of $A(1, -2)$ and $B(4, 7)$ externally in the ratio of $2 : 1.$

  1. $(7, 16)$
  2. $(7,12)$
  3. $\left(3,\displaystyle \frac{16}{3}\right)$
  4. $(3,16)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
The given points are $A(1,-2)$ and $B(4,7)$. 
We have to find the coordinate of points which divide the line segment externally in the ratio $2:1$
Let point $C$ divides the segment $AB$ in the ratio $1:2$, hence $m=2, n=1$
By section formula which states that when the line segment is divided externally by the point in the ration $m:n$ then coordinates of point are

$\Rightarrow$  $C=\left(\dfrac{mx _2-nx _1}{m-n},\,\dfrac{my _2-n _1}{m-n}\right)$

           $=\left(\dfrac{(2)(4)-(1)(1)}{2-1},\,\dfrac{(2)(7)-(1)(-2)}{2-1}\right)$

           $=\left(\dfrac{7}{1},\dfrac{16}{1}\right)$

           $=(7,16)$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The point (11, 10) divides the line segment joining the points (5, -2) and (9, 6) in the ratio

  1. 1 : 3 internally

  2. 1 : 3 externally

  3. 3 : 1 internally

  4. 3 : 1 externally
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the section formula, if a point $(x,y)$ divides the line joining the points $({ x } _{ 1 },{ y } _{ 1

})$ and $({ x } _{ 2 },{ y } _{ 2 })$ in the ratio $ m:n $, then $(x,y) =

\left( \dfrac { m{ x } _{ 2 } + n{ x } _{ 1 } }{ m + n } ,\dfrac { m{ y } _{ 2

}  + n{ y } _{ 1 } }{ m + n }  \right) $

Let the ratio be $ k : 1 $


Substituting $({ x } _{ 1 },{ y } _{

1 }) = (5,-2) $ and $({x } _{ 2 },{ y } _{ 2 }) = (9,6) $  in the

section formula, we get  $ \left( \dfrac { k(9)  + 1(5) }{ k + 1 }

,\dfrac { k(6) + 1(-2) }{ k + 1 }  \right) = ( 11,10) $ 


$ \left( \dfrac { 9k + 5}{ k + 1 }

,\dfrac { 6k - 2 }{ k + 1} \right) = ( 11,10) $


Comparing the x - coordinate,

$ => \dfrac { 9k + 5 }{ k + 1 } = 11 $

$ =>9k + 5 = 11k + 11 $


$ 2k = -6 $


$ k = -3 $

Hence, the ratio is $ 3:1$ externally.

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

Value of m for which the point P(m, 6) divides the join of A(-4, 3) and B(2, 8) is

  1. 5
  2. $\displaystyle \frac{3}{2}$
  3. $\displaystyle -\frac{2}{5}$

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

Equation

of a line joining two points $ { (x } _{ 1 },{ y } _{ 1 }) $ and $ { (x } _{ 2

},{ y } _{ 2 }) $ is given by the formula $ y-{ y } _{ 1 }=\quad \left( \dfrac {

{ y } _{ 2 }-{ y } _{ 1 } }{ { x } _{ 2 }-{ x } _{ 1 } }  \right) (x-{ x } _{ 1

}) $

Equation of line passing through A $(-4,3)

$ and B $ (2,8)$  is  $ y-3=\quad \left( \dfrac { 8-3 }{ 2+4 }  \right) (x+4) $
$

=> y-3=\dfrac { 5 }{ 6 } (x+4) $

$ => 6y-18=5x+20$

$ => 5x-6y+38=0$

Since Point $  P(m, 6)  $ divides this line, it should satisfy the equation of the line, if we substitute

$ x = m $ and $ y =6 $ in it.


So, $  5m-36+38=0 $

$ => 5m = -2 $

$ m = -\dfrac {2}{5} $

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment
Find the coordinates of the point which divides the line segment joining the points (6, 3) and (-4, 5) in the ratio 3 : 2 externally.

  1. (-12, 9)

  2. (-16, 9)

  3. (-24, 9)

  4. (-14, 9)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the section formula, if a point $(x,y)$ divides the line joining the points

$({ x } _{ 1 },{ y } _{ 1 })$ and $({ x } _{ 2 },{ y } _{ 2 })$externally  in the ratio $ m:n $, then $(x,y) = \left(

\dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n } ,\dfrac { m{ y } _{ 2 }-n{ y } _{ 1 }

}{ m-n }  \right) $


Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (6,3) $ and

$({x } _{ 2 },{ y } _{ 2 }) = (-4,5) $  and $ m = 3, n = 2 $ in the section formula, we get 



$ C = \left( \dfrac { 3(-4)-2(6) }{ 3-2 } ,\dfrac { 3(5)-2(3) }{ 3-2 } 

\right) =\left( -24,9 \right) $

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

If the line joining A(2, 3) and B(-5, 7) is cut by x-axis at P then AP : PB is

  1. 3 : 7

  2. -3 : 7

  3. 7 : 3

  4. 7 : -3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the section formula, if a

point $(x,y)$ divides the line joining the points $({ x } _{ 1 },{ y } _{ 1

})$ and $({ x } _{ 2 },{ y } _{ 2 })$ in the ratio $ m:n $, then $(x,y) =

\left( \dfrac { m{ x } _{ 2 } + n{ x } _{ 1 } }{ m + n } ,\dfrac { m{ y } _{ 2

}  + n{ y } _{ 1 } }{ m + n }  \right) $


Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (2,3) $ and $({x } _{ 2 },{ y } _{ 2

}) = (-5,7) $  in the section formula, we get the point $ \left( \dfrac {

m(-5)  + n(2) }{ m + n } ,\dfrac { m(7) + n(3) }{ m + n }  \right)

=\left( \dfrac { -5m  + 2n }{ m + n } ,\dfrac { 7m + 3n }{ m + n} \right) $


As the point lies on x - axis, y -coordinate $ = 0 $.

$ => \dfrac { 7m + 3n }{ m + n} = 0 $ 

$ => 7m = -3n $  or $ m : n = -3:7 $

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

Find the coordinates of the point which divides the line segment joining the points $(6, 3)$ and $(-4, 5)$ in the ratio $3 : 2$, externally.

  1. $(24,9)$
  2. $(-24,-9)$
  3. $(-24,9)$
  4. $(24,-9)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let P$(x,y)$ be the required point.
Using the section formula, if a point $(x,y)$ divides the line joining the points $({ x } _{ 1 },{ y } _{ 1 })$ and $({ x } _{ 2 },{ y } _{ 2 })$externally  in the ratio $ m:n $, then $(x,y) = \left( \dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n } ,\dfrac { m{ y } _{ 2 }-n{ y } _{ 1 } }{ m-n }  \right) $
Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (6,3) $ and $({x } _{ 2 },{ y } _{ 2 }) = (-4,5) $  and $ m = 3, n = 2 $ in the section formula, we get 

$ P = \left( \dfrac { 3(-4)-2(6) }{ 3-2 } ,\dfrac { 3(5)-2(3) }{ 3-2 } \right) =\left(-24,9 \right) $

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

The ordinate of the point which divides the line joining the origin and the point (1, 2) externally in the ratio of 3 : 2 is

  1. $-2$
  2. $\displaystyle\frac{3}{5}$
  3. $\displaystyle\frac{2}{5}$
  4. $6$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The co-ordinates of the required point will be

$\displaystyle y=\dfrac{m _1y _2-m _2y _1}{m _1-m _2}$

$\displaystyle=\dfrac{3\times2-2\times0}{3-2}=6$

Multiple choice maths transformations midpoint of line segment mid point formula mid-point of a line segment

Find the coordinates of the point which divides the join of the points $(2,4)$ and $(6,8)$ externally in the ratio $5:3$.

  1. $(12,14)$
  2. $(14,12)$
  3. $(-12,14)$
  4. $(12,-14)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $A(2, 4)$ and $B(6,8)$

Applying the section formula externally,

$\left( \dfrac { L{ x } _{ 2 }-{ mx } _{ 1 } }{ L-m } ,\dfrac { L{ y } _{ 2 }-{ my } _{ 1 } }{ L-m }  \right)$

Here the ratio given is $5:3$ that is $L=5$ and $m=3$, therefore,

$\left( \dfrac { L{ x } _{ 2 }-{ mx } _{ 1 } }{ L-m } ,\dfrac { L{ y } _{ 2 }-{ my } _{ 1 } }{ L-m }  \right) =\left( \dfrac { (5\times 6)-(3\times 2) }{ 5-3 } ,\dfrac { (5\times 8)-(3\times 4) }{ 5-3 }  \right) $


$=\left( \dfrac { 30-6 }{ 2 } ,\dfrac { 40-12 }{ 2 }  \right) =\left( \dfrac { 24 }{ 2 } ,\dfrac { 28 }{ 2 }  \right) =\left( 12,14 \right)$ 

Hence, the coordinates of the point is $(12,14)$.