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.

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Coordinate Geometry and Construction Questions

Multiple choice maths constructions mid-point formula midpoints division of a line segment

A (a,b) and (0,0) are two fixed points, ${ M } _{ 1 }$ is the mid points of AB, ${ M } _{ 2 }$ is the midpoint of $A{ M } _{ 1 },{ M } _{ 3 }$ is the midpoint of $A{ M } _{ 2 }$ and so on then ${ M } _{ 5 }$ =in

  1. $\left( \dfrac { 7a }{ 8 } ,\dfrac { 7b }{ 8 } \right) $
  2. $\left( \dfrac { 15a }{ 16 } ,\dfrac { 15b }{ 16 } \right) $
  3. $\left( \dfrac { 31a }{ 32 } ,\dfrac { 15b }{ 32 } \right) $
  4. $\left( \dfrac { 63a }{ 64 } ,\dfrac { 15b }{ 64 } \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $A(a, b)$ and $B(0, 0)$ are two fixed points. $M _1$ is the mid point of $\overline{AB}$, $M _2$ is the mid point of $\overline{AM _1}$, $M _3$ is the mid point of $\overline{AM _2}$ and so on, then $M _5$ is?

  1. $\left(\dfrac{7a}{8}, \dfrac{7b}{8}\right)$
  2. $\left(\dfrac{15a}{16}, \dfrac{15b}{16}\right)$
  3. $\left(\dfrac{31a}{32}, \dfrac{31b}{32}\right)$
  4. $\left(\dfrac{63a}{64}, \dfrac{63b}{64}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths constructions mid-point formula midpoints division of a line segment

The co-ordinates of the mid point of segment $KR$, where $K(2.5, -4.3)$ and $R(-1.5, 2.7)$, are

  1. $(0.5, 0.8)$
  2. $(-0.5, -0.8)$
  3. $(-0.5, 0.8)$
  4. $(0.5, -0.8)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given that $KR$ is a line segment joining the points $K=(2.5, -4.3)$ and $R=(-1.5, 2.7)$ 

We know that the co-ordinate of the midpoint $M$ of a line joining the points $(x _1, y _1)$ and $(x _2,y _2)$ is given by 

$M=\left( \dfrac { { x } _{ 1 }+{ x } _{ 2 } }{ 2 }, \dfrac { { y } _{ 1 }+{ y } _{ 2 } }{ 2 }  \right)$
Let $M _{KR}$ be the midpoint of the line segment $KR$

 ${ \therefore \quad M } _{ KR }=\left( \dfrac {2.5-1.5}{2}, \dfrac{-4.3+2.7}{2}  \right) =(0.5,-0.8)$

Hence, option D is correct.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Mid-point of the line-segment joining the points $(-5,4)$ and $(9, -8)$ is:

  1. $(-7,6)$
  2. $(2, -2)$
  3. $(7,-6)$
  4. $(-2, -2)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Midpoint of two points $ { (x } _{ 1 },{ y } _{ 1 }) $ and $ { (x } _{ 2 },{ y } _{
2 }) $ is calculated by the formula $ \left( \cfrac { { x } _{ 1 }+{ x} _{ 2 } }{ 2 } ,\cfrac { { y } _{ 1 }+y _{ 2 } }{ 2 }  \right) $
Using this formula, mid - point of the line-segment joining the points $(5,4)$ and $(9,8)$ is: $= \left( \cfrac { -5 + 9 }{ 2 } ,\cfrac { 4- 8 }{ 2 }  \right)  = (2,-2) $

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The mid-point of a line segment is $(5,8)$. If one end point is $(3,5)$, find the second end point

  1. $(-7,11)$
  2. $(7,-11)$
  3. $(7,11)$
  4. $(-7,-11)$
Reveal answer Fill a bubble to check yourself
A,C Correct answer
Explanation

Mid point of two points $ { (x } _{ 1 },{ y } _{ 1 }) $ and $ { (x } _{ 2 },{ y } _{

2 }) $ is calculated by the formula $ \left( \dfrac { { x } _{ 1 }+{ x

} _{ 2 } }{ 2 } ,\dfrac { { y } _{ 1 }+y _{ 2 } }{ 2 }  \right) $

Let the other point be $ (x,y) $
So, $=\left( \dfrac {3 + x }{ 2 } ,\dfrac { 5 + y }{ 2 }  \right) \quad =\quad (5,8)

$
$ => \frac {3 + x }{ 2 } = 5 $ and  $ \dfrac { 5 + y } {2} = 8 $
$ => x = 7 , y = 11 $
So, the second end point is $ (7,11) $

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the mid-point of AB where A and B are the points $(-5, 11)$ and $(7,3)$, respectively.

  1. $(1,7)$
  2. $(0,0)$
  3. $(1,0)$
  4. $(0,7)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Midpoint of two points $ { (x } _{ 1 },{ y } _{ 1 }) $ and $ { (x } _{ 2 },{ y } _{

2 }) $ is  calculated by the formula $ \left( \dfrac { { x } _{ 1 }+{ x

} _{ 2 } }{ 2 } ,\dfrac { { y } _{ 1 }+y _{ 2 } }{ 2 }  \right) $

Using this formula, mid point of AB $= \left( \dfrac { -5 + 7 }{ 2 } ,\dfrac { 11 + 3 }{ 2 } 

\right) = (1, 7) $

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The mid-point of a line is $(-4,-2)$ and one end of the line is $(-6,4)$. The co-ordinates of the other end are

  1. $(2,-8)$
  2. $(-2,8)$
  3. $(-2,-8)$
  4. $(2,8)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The co-ordinate of the mid point of a line segment whose end points are $(x _1,y _) $ and $(x _2,y _2)$ are given by $(\cfrac{x _1+x _2}{2},\cfrac{y _1+y _2}{2})$

Substituting $(x _1,y _1)=(-6,4)$ and mid-point $(-4,-2)$.
$\Rightarrow (-4,-2)=(\cfrac{-6+x _2}{2},\cfrac{4+y _2}{2})$
Equating the $x-$coordinates
$\Rightarrow -4=\cfrac{-6+x _2}{2}$
$\Rightarrow -4\times 2=-6+x _2$
$\Rightarrow x _2=6-8$
$\Rightarrow x _2=-2$
Equating the $y-$coordinates

$\Rightarrow -2=\cfrac{4+y _2}{2}$
$\Rightarrow -2\times 2=4+y _2$
$\Rightarrow -y _2=4+4$
$\Rightarrow y _2=-8$

The coordinates of the other end are $(-2,-8)$.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Calculate mid point of $A(5,\,3)$ and $B(9,\,8)$

  1. $\dfrac{11}{2},\,7$
  2. $7,\,\dfrac{11}{2}$
  3. $7,\,11$
  4. $14,\,11$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The mid point of line segment joining $(x _1,\,y _1)$ and $(x _2,\,y _2)$ is $\begin{pmatrix}\dfrac{x _1+x _2}{2},\,\dfrac{y _1+y _2}{2}\end{pmatrix}$
Mid point of AB $\Rightarrow\begin{pmatrix}\dfrac{5+9}{2},\,\dfrac{3+8}{2}\end{pmatrix}$
$\Rightarrow\begin{pmatrix}7,\,\dfrac{11}{2}\end{pmatrix}$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the midpoint between the coordinates $(9, 3)$ and $(1, 1)$.

  1. $\left(5, 2\right)$
  2. $\left(3, 2\right)$
  3. $\left(5, 1\right)$
  4. $\left(3, 1\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We know the midpoint formula between two points.
$\left(\dfrac{x _{1}+x _{2}}{2},\dfrac{y _{1}+y _{2}}{2}\right)$
Substitute the values, we get
$=$ $\left(\dfrac{9+1}{2},\dfrac{3+1}{2}\right)$
$=$ $\left(\dfrac{10}{2},\dfrac{4}{2}\right)$
Therefore, midpoint is $\left(5, 2\right)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the value of $k$, so that $(2, 1)$ is the midpoint between $(1, k)$ and $(3, 1)$.

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

We know the midpoint formula between two points.
$\left(\dfrac{x _{1}+x _{2}}{2},\dfrac{y _{1}+y _{2}}{2}\right)$
Substitute the values, we get
$\left(\dfrac{1+3}{2},\dfrac{k+1}{2}\right)$ $=$ $\left(2, 1\right)$
On equating, we get
$\left(\dfrac{k+1}{2}\right)$ $= 1$
$\Rightarrow k + 1 = 2$
$\Rightarrow k = 2 - 1$
$\Rightarrow k = 1$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

In the standard $(x,y)$ coordinate plane, what are the coordinates of the midpoint of a line segment whose endpoints are $(-3,0)$ amd $(7,4)$?

  1. $(2,2)$
  2. $(2,3)$
  3. $(5,2)$
  4. $(5,4)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given two points $(x _{1},y _{1})$ and $(x _{2},y _{2})$, then the coordinates of the midpoint are $\left (\dfrac{x _{1}+x _{2}}{2},\dfrac{y _{1}+y _{2}}{2}\right)$.
In the above case the points are $(-3,0)$ and $(7,4)$. 

Hence, the midpoint is $\left (\dfrac{-3+7}{2},\dfrac{0+4}{2}\right)=(2,2)$.

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Points $A(\sqrt {2}, 4), B(6, -\sqrt {3})$ and $C$ are collinear. If $B$ is the midpoint of line segment $AC$, approximately calculate the $(x, y)$ coordinates of point $C$.

  1. $(3.71, 1.13)$
  2. $(3.71, 5.73)$
  3. $(7.41, -7.46)$
  4. $(10.59, -7.46)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given points $A$ $(\sqrt{2},4)$, $B$ $(6,-\sqrt{3})$ and $C$ are col-linear if $B$ the mid point $AC$. 

 if (A) $(3.71,1.13)$ is point $C$, then point $(x,y) =$ $\dfrac{\sqrt{2}+3.71}{2},\dfrac{4+1.13}{2}$
$\Rightarrow 2.56,2.56$ this not point $B$ 

 if (B) $(3.71,5.73)$ is point $C$, then point $(x,y) =$ $\dfrac{\sqrt{2}+3.71}{2},\dfrac{4+5.73}{2}$
$\Rightarrow 2.56,4.86$ this not point $B$ 

 if (C) $(7.41,-7.46)$ is point $C$, then point $(x,y) =$ $\dfrac{\sqrt{2}+7.41}{2},\dfrac{4-7.46}{2}$
$\Rightarrow 4.41,-1.73$ this not point $B$ 
 if (D) $(10.59,-7.46)$ is point $C$, then point $(x,y) =$ $\dfrac{\sqrt{2}+10.59}{2},\dfrac{4-7.46}{2}$
$\Rightarrow 6,1.73$ this the  point $B (6,1.73)$
 if (E) $(10.59,5.73)$ is point $C$, then point $(x,y) =$ $\dfrac{\sqrt{2}+10.59}{2},\dfrac{4+5.73}{2}$
$\Rightarrow 6,4.86$ this not point $B$. 
So (D) $(10.59,-7.46)$ is mid point $B$ of line $AC$.