Mathematics · Quantitative Aptitude

Coordinate Geometry and Construction

104 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

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 $AC$.

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

Given point $A(-3, -8)$, if the midpoint of segment $AB$ is $(1, -5)$, calculate the coordinates of point $B$.

  1. $(5, -2)$
  2. $(4, -2)$
  3. $(-1, -6.5)$
  4. $(-2, -2)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, coordinates $A (-3,-8)$ and the mid point of $AB$ is $(1,-5)$

As per Midpoint formula coordinates of mid point $=$ $\dfrac{x _{1}+x _{2}}{2},\dfrac{y _{1}+y _{2}}{2}$
Let the  coordinates of point $B$ be $(x,y)$
Then $(1,-5)=\left [ \left (\dfrac{-3+x}{2}\right),\left (\dfrac{-8+y}{2}\right) \right ]$
Then $\dfrac{-3+x}{2}=1$
$\Rightarrow -3+x=2$
$\Rightarrow x=2+3=5$
Then$\dfrac{-8+y}{2}=-5$
$\Rightarrow -8+y=-10$
$\Rightarrow y=-10+8=-2$
Then coordinates of $B$ is $(5,-2)$.

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

If $\left (\dfrac {a}{3}, 4\right )$ is the midpoint of the line segment joining $A (-6, 5)$ and $B(-2, 3)$, find $a$.

  1. $-4$
  2. $-12$
  3. $12$
  4. $-6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the point A(-6,5) and B(-2,3) $\left( \dfrac { a }{ 3 } ,4 \right) $ is the middle of AB

$\Rightarrow \left( \dfrac { -2-6 }{ 2 } ,\dfrac { 3+5 }{ 2 }  \right) =\left( \dfrac { a }{ 3 } ,4 \right) \ \Rightarrow \dfrac { -8 }{ 2 } =\dfrac { a }{ 3 } \ \therefore a=-\dfrac { 24 }{ 2 } =-12\ $

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

$A(-3,2)$ and $B(5,4)$ are the end points of a line segment, find the coordinates of the midpoints of the line segment.

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

Since $A(-3,2)\equiv(x _1,y _1)$ and $B(5,4)\equiv(x _2,y _2)$ are the end points of a line segment.


Therefore, the coordinates of the midpoints of the line segment is given by:


$(x,y)=\left( \dfrac { { x } _{ 1 }+{ x } _{ 2 } }{ 2 } ,\dfrac { { y } _{ 1 }+{ y } _{ 2 } }{ 2 }  \right)$

$ \\ \Rightarrow (x,y)=\left( \dfrac { -3+5 }{ 2 } ,\dfrac { 2+4 }{ 2 }  \right) \quad $

$\\ \Rightarrow (x,y)=\left( \dfrac { 2 }{ 2 } ,\dfrac { 6 }{ 2 }  \right) =\left( 1,3 \right)$ 

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

Find the midpoint of the line segment joining the points  $(1,-1)$ and $(-5,-3)$

  1. $(-2,1)$
  2. $(2,1)$
  3. $(-2,-1)$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Take $(x _1,y _1)=(1,-1)$ and $(x _2,y _2)=(-5,-3)$.

By midpoint theorem:

$x=\dfrac{x _1+x _2}{2}$ and $y=\dfrac{y _1+y _2}{2}$

Hence, $x=\dfrac{1+(-5)}{2}=-2$ and $y=\dfrac{-1-(-3)}{2}=1$.

So, $(x,y)=(-2,1)$.
Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the mid point of (3,8) and (9,4).

  1. $(5,6)$
  2. $(6,6)$
  3. $(4,4)$

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

Midpoint formula is given by $\left(\dfrac{x _1+x _2}{2},\dfrac{y _1+y _2}{2} \right)$

So midpoint of $(3,8)$ and $(9,4)$ is $=\left(\dfrac{3+9}{2},\dfrac{8+4}{2} \right)=(6,6)$

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

Find the mid point of $(4,6)$ and $(2,-6)$.

  1. $(3,4)$
  2. $(2,-2)$
  3. $(3,0)$

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

We know that end points of a line segment is $(a,b)$ and $(c,d)$, then the midpoint of the line segment has the coordinates:

$\dfrac{a+c}{2},\dfrac{b+d}{2}$
Then mid point of line segment $(4,6)$ and $(2,-6)$ is 

$\dfrac{4+2}{2},\dfrac{6-6}{2}$
$\Rightarrow \dfrac{6}{2},\dfrac{0}{2}$
$\Rightarrow (3,0)$

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

If mid point of the line segment joining (2a, 4) and (-2, 3b) is  (1, 2a + 1), then the values of  a and b are given by

  1. $a = 2, b = - 2$
  2. $a = b = 2$
  3. $a= 1 = b$
  4. $a= -2, b = 2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Midpoint of any two points $(a.b)$ and $(c,d)$ is given by
$M=\left(\dfrac{a+c}{2},\dfrac{b+d}{2}\right)$
Given points are $(2a, 4)$ and $(-2,3b)$ 
$\therefore M=\left(\dfrac{2a-2}{2},\dfrac{4+3b}{2}\right)$
$\implies (1,2a+1)=\left(a-1,\dfrac{4+3b}{2}\right)$
$\implies a-1=1$ and $\dfrac{4+3b}{2}=2a+1$
$\implies a=2$ and $4a-3b=2$
$\implies a=2, b=2$
Hence, $a=b=2$.
Multiple choice maths constructions mid-point formula midpoints division of a line segment

If $P \left( \dfrac{a}{3},\dfrac{b}{2} \right)$ is the mid-point of the line segment joining $A(-4,3)$ and $B(-2,4)$ then $(a,b)$ is 

  1. $(-9,7)$
  2. $\left( -3, \dfrac{7}{2} \right)$
  3. $(9,-7)$
  4. $\left( 3, -\dfrac{7}{2} \right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Using the midpoint formula, $\left(x,y\right)=\left(\dfrac{{x} _{1}+{x} _{2}}{2},\dfrac{{y} _{1}+{y} _{2}}{2}\right)$ where ${x} _{1}=-4$,${y} _{1}=3$,${x} _{2}=-2$, ${y} _{2}=4$
Given $P$ is a mid-point of $AB$
$\Rightarrow P\left(\dfrac{a}{3},\dfrac{b}{2}\right)=\left(\dfrac{-4-2}{2},\dfrac{3+4}{2}\right)=\left(-3,\dfrac{7}{2}\right)$
Equating the $x$ and $y$ coordinates, we get
$\Rightarrow \dfrac{a}{3}=-3,\dfrac{b}{2}=\dfrac{7}{2}$
$\Rightarrow a=-9,b=7$
$\therefore \left(a,b\right)=\left(-9,7\right)$

Multiple choice

What is the name of the algorithm that finds the intersection of two line segments?

  1. Line-segment intersection algorithm

  2. Point-line intersection algorithm

  3. Line-plane intersection algorithm

  4. Plane-plane intersection algorithm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The line-segment intersection algorithm is an algorithm that finds the intersection of two line segments in $O(1)$ time.

Multiple choice

How many segments does the Divided Line consist of?

  1. Two

  2. Three

  3. Four

  4. Five
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Divided Line consists of four segments, each representing a different level of understanding.