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.

Midpoint calculationsSection formulas3D geometry projectionsGeometric construction stepsLine segment ratios

Coordinate Geometry and Construction Questions

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

$M(2, 6)$ is the midpoint of $\overline {AB}$. If $A$ has coordinates $(10, 12)$, the coordinates of $B$ are

  1. $(6, 10)$
  2. $(-6, 0)$
  3. $(-8, -4)$
  4. $(18, 16)$
  5. $(22, 18)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Mid point $=$ $\dfrac{x _1+x _2}{2}, \dfrac{y _1+y _2}{2}$

Here mid-point $=(2,6)$
$x _1=10,y _1=12$
$\Rightarrow (2,6)=\dfrac{10+x _2}{2},\dfrac{12+y _2}{2}$
$\Rightarrow \dfrac{10+x _2}{2}=2; \dfrac{12+y _2}{2}=6$
$\Rightarrow x _2=4-10;y _2=12-12$
$\Rightarrow x _2=-6,y _2=0$
$\therefore $ co-ordinates of $B=(-6,0)$.

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

If a point $C$ be the mid-point of a line segment $AB$, then $AC = BC = (...) AB$.

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

If $C$ is the midpoint of $AB$, then $C$ divides $AB$ in equal segments. Those segments are $AC$ and $AB$.

therefore, $AC=BC$. 
 $AC+BC=AB$ (As AC and BC are the segments of $AB$)
$\Rightarrow 2AC=2BC=AB\ \Rightarrow AC=BC=\dfrac{1}{2} AB$

Multiple choice maths constructions mid-point formula midpoints division of a line segment
Say true or false.
The mid-point of the line segment joining the points $P(x _1, y _1)$ and $Q(x _2, y _2)$ is 
$\dfrac {x _1+x _2}{2}, \dfrac {y _1+y _2}{2}.$

  1. True

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

The midpoint formula for a line segment with endpoints (x1, y1) and (x2, y2) is indeed ((x1+x2)/2, (y1+y2)/2). This is a standard coordinate geometry definition.

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

The mid point of line $AB$ with $A(2,3)$ and $B(5,6)$

  1. $(3.5,4.5)$
  2. $(3,4)$
  3. $(4,5)$

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

Given points $A(2,3)\equiv(x _1,y _1)$ and $B(5,6)\equiv(x _2,y _2)$


Mid points given as ,


$\Rightarrow\left(\dfrac{x _1+x _2}2,\dfrac{y _1+y _2}2\right)$

$\Rightarrow\left(\dfrac{2+5}2,\dfrac{3+6}2\right)$

$\Rightarrow\left(3.5,4.5\right)$

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

The mid-point of the line $(a, 2)$ and $(3, 6)$ is $(2, b)$. Find the numerical values of $a$ and $b$.

  1. $a=1$, $b=6$
  2. $a=2$, $b=4$
  3. $a=1$, $b=4$
  4. $a=2$, $b=6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mid-point of $(a,2)$ and $(3,6)$ is $(2,b)$

=>$(2,b)=\left( \cfrac { a+3 }{ 2 } ,\cfrac { 2+6 }{ 2 }  \right) \ =>a=4-3,b=4\ =>a=1,b=4$

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

The co -ordinates of the midpoint of a line segment joining $ p(5,7) $ and $ Q (-3,3) $ are........

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

$P(5,7),Q(-3,3)$

mid point is given by,

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

$\Rightarrow (x,y)=(1,5)$
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
C Correct answer
Explanation

Let A be (a, b) and B be (0, 0). The midpoint formula shows that each successive midpoint moves closer to A by a factor of half the remaining distance. Specifically, M1 = (a/2, b/2), M2 = a - (a - M1)/2, and following the geometric progression for n steps, the coordinate for Mn is given by (a(1 - 1/2^n), b(1 - 1/2^n)). For M5, substituting n = 5 gives 1 - 1/32 = 31/32, resulting in (31a/32, 31b/32).

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 $\Rightarrow\begin{pmatrix}\dfrac{5+9}{2},\,\dfrac{3+8}{2}\end{pmatrix}$
$\Rightarrow\begin{pmatrix}7,\,\dfrac{11}{2}\end{pmatrix}$