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 constructions mid-point formula midpoints division of a line segment

Find the mid point of $(9,5)$ and $(3,7)$

  1. $(6,6)$
  2. $(12,12)$
  3. $(2,2)$
  4. $(1,1)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given points $(9,5),(3,7)$

Mid point is given as $\left(\dfrac{x _1+x _2}2,\dfrac{y _1,y _2}{2}\right)\\left(\dfrac{9+3}{2},\dfrac{5+7}{2}\right)\\left(\dfrac{12}{2},\dfrac{12}{2}\right)=(6,6)$

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

The mid-point of the line segment joining $( 2a, 4)$ and $(-2, 2b)$ is $(1, 2a + 1 )$. The values of $a$ and $b$ are 

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

Midpoint of two points $ =\left( \cfrac { { x } _{ 1 }+{ x} _{ 2 } }{ 2 } ,\cfrac { { y } _{ 1 }+y _{ 2 } }{ 2 }  \right) $
Given, midpoint of $ (2a,4) $ and $ (-2,2b) = (1,2a+1) $
$ => \left(\cfrac { 2a-2 }{ 2 } ,\cfrac { 4+2b }{ 2 }\right)= (1,2a+1) $
$ => \cfrac { 2a-2 }{ 2 } = 1 ; \cfrac { 4+2b }{ 2 } = 2a + 1 $
$ => 2a -2 = 2 $
$=> a = 2 $

And, $ \cfrac { 4+2b }{ 2 } = 2a + 1 $
$=> \cfrac { 4+2b }{ 2 } = 2(2) + 1 = 5 $
$ => 4 + 2b = 10 $
$ => 2b = 6 $
$=> b = 3 $

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

The point which lies in the perpendicular bisector of the line segment joining the points A (-2, -5)  and B (2,5) is 

  1. (0, 0)

  2. (0, 2)

  3. (2, 0)

  4. (-2, 0)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A perpendicular bisector of a line segment, passed through its midpoint.

If C is the point on AB, through which its perpendicular bisector passes, then C $ = $ mid point of AB.

Mid

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

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

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





Using this formula,


mid point of AB $= \left( \frac { -2+2 }{ 2 } ,\frac { -5+5 }{ 2 }  \right)

\quad =\quad (0,0) $



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

If Q$\displaystyle \left ( \frac{a}{3},4 \right )$ is the mid-point of the line segment joining the points A(-6,5) and B(-2,3), then the value of 'a' is

  1. 4

  2. -6

  3. -8

  4. -12
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The co-ordinates of the mid-point  of the line segment joining the point $P(x _1,y _1),Q(x _2,y _2)$$=\left(\dfrac{x _1+x _2}{2},\dfrac{y _1+y _2}{2}\right)$

$\Rightarrow \left(\dfrac{a}{3},4\right)=\left(\dfrac{-6+(-2)}{2},\dfrac{5+3}{2}\right)$
$\Rightarrow \left(\dfrac{a}{3},4\right)=\left(\dfrac{-8}{2},4\right)$
X co-ordinates
$\Rightarrow \dfrac{a}{3}=\dfrac{-8}{2}$
$\Rightarrow a=\dfrac{-8\times 3}{2}=-12$

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

The midpoint of the line segment between P$\displaystyle _{1}$ (x, y) and P$\displaystyle _{2}$ (-2, 4) is P$\displaystyle _{m}$ (2, -1). Find the coordinate.

  1. (6, -5)

  2. (5, -6)

  3. (6, -6)

  4. (-6, 6)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given,

$P _m(2,-1), P _1(x,y)$ and $P _2(-2,4)$

$(2,-1)=\left(\dfrac{x-2}{2}, \dfrac{y+4}{2}\right)$   ..... By midpoint formula 

$\therefore 2=\dfrac{x-2}{2}$
$=>4=x-2$
$=>x=6$

And,
$-1=\dfrac{y+4}{2}$
$=>-2=y+4$
$=>y=-6$

$\therefore (x,y)=(6,-6)$

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

In the xy-plane, find the mid point of the line segment joining the points $\left( 5,9 \right) $ and $\left( 7,11 \right) $.

  1. $(1.5, 2)$
  2. $(6, 10)$
  3. $(5.5, 5)$
  4. $(6, -3.5)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given line segment point $(5,9)$ and $(7,11)$, then mid point as per section formula: 

$(x,y)=$ $\left ( \dfrac{7+5}{2} ,\dfrac{11+9}{2} \right )$
$=$ $\left ( \dfrac{12}{2} ,\dfrac{20}{2}\right)$
$=$ $ (6,10)$

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

$M$ is the midpoint of the straight line $PQ$. If $P(-2,9)$ and $M$ is $(4,3)$, find the coordinates of $Q$.

  1. $(1,6)$
  2. $(10,-3)$
  3. $(10,6)$
  4. $(8,-3)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\dfrac{x-2}{2}=4$ and $\dfrac{y+9}{2}=3$
 $\Rightarrow x=10$ and $y=-3$ 
           $a(10,-3)$
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 the mid-point between the points $(a+ b, a- b)$ and $(-a, b)$ lies on the line $ax + by = k$, what is k equal to?

  1. $\dfrac ab$
  2. $a + b$
  3. $ab$
  4. $a - b$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mid point of points $(a+b,a-b),(-a,b)$ is $\left( \dfrac { a+b-a }{ 2 } ,\dfrac { a-b+b }{ 2 }  \right) =\left(\dfrac { b }{ 2 } ,\dfrac { a }{ 2 } \right)$

It lies on line $ax+by=k$ then 
$\Rightarrow a\times \dfrac{b}{2}+b\times \dfrac{a}{2}=k$
$\Rightarrow ab=k$

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,

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

$\Rightarrow (x,y)=(1,5)$