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 section and mid-point formula midpoint of line segment mid point formula mid-point of a line segment

STATEMENT - 1 : The coordinates of the point P(x, y) which divides the line segment joining the points A$(x _1,  y _1)$ and B$(x _2,  y _2)$ internally in the ration $m _1$  :  $m _2$ are $\left ( \dfrac{m _1 x _2 -m _2 x _1}{m _1 + m _2} ,  \dfrac{m _1 y _2 - m _2 y _1}{m _1 + m _2}\right )$


STATEMENT - 2 : The mid-point of the line segment joining the points P $(p _1 y _1)$ and Q$(x _2, y _2)$ is $\left ( \dfrac{x _1+x _2}{2} , \dfrac{y _1 + y _2}{2} \right )$

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Statement -1 is false,

As the formula is not for the internally it is when point divides externally.
Statement -2 is true.

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

Consider points $A(-1,3), B(-1,2)$. Find point $P$ which divides $AB$ externally in $\dfrac{5}{4}$.

  1. $(9,-22)$
  2. $(-1,2)$
  3. $(-1,-2)$
  4. $(9,22)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let point P be (x,y)
$x=\cfrac { 5\times (-1)-4\times (-1) }{ 5-4 } \\ x=\cfrac { -5+4 }{ 1 } \\ x=-1\\ y=\cfrac { 5\times (2)-4\times (3) }{ 5-4 } \\ y=\cfrac { 10-12 }{ 1 } \\ y=-2\\ \therefore P=(-1,-2)$
Multiple choice maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

Mid-point theorem states that:

  1. The line segment joining the mid-points of two sides of a triangle is not parallel to the third side and equal to half the length of the third side.

  2. The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to one-third the length of the third side.

  3. The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to the length of the third side.

  4. The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Mid-point theorem states that

The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
Hence, option D is correct.

Multiple choice maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

Fill in the blanks:
(i) The ling segment joining a vertex of a triangle to the midpoint of its opposite side is called a $\underline { P } $ of the triangle.
(ii) The perpendicular line segment from a vertex of a triangle to its opposite is called an $\underline { Q } $ of the triangle
(iii) A triangle has $\underline { R } $ altitudes and $\underline { S } $ medians

  1. $P-$ Altitude; $Q$- Median; $R-1$; $S-1$
  2. $P-$ Altitude; $Q$- Median; $R-3$; $S-3$
  3. $P-$ Median; $Q$- Altitude; $R-3$; $S-3$
  4. $P-$ Median; $Q$- Altitude; $R-2$; $S-3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Medians : The line segment from any vertex of a triangle to the midpoint of its opposite side is called medians of triangle. 

Altitude : The perpendicular drawn from a vertex to opposite side is called as altitude. 
A triangle has $3$ altitudes & $3$ medians.

Multiple choice maths mid-point and its converse proving the mid-point theorem the mid-point theorem mid point theorem

M is the midpoint of $\displaystyle\overline{AB}$. The coordinates of A are $(-2,3)$ and the coordinates of M are $(1,0)$. Find the coordinates of B.

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

  5. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $(x,y)$ be the coordinates of B.
According to question, M is mid point of A and B.
$\Rightarrow \dfrac{-2+x}2=1\Rightarrow x=4$
and $ \dfrac{3+y}2=0\Rightarrow y=-3$
Therefore B is $(4,-3)$
Option B is correct.

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

If $O(0,4)$ and $P(0,-4)$, are the co-ordinates of the line segment $OP$ then co-ordinate of its midpoint are

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

Midpoint of a line segment having coordiantes $\left({x} _{1},{y} _{1}\right)$ and $\left({x} _{2},{y} _{2}\right)$ is $\left(\dfrac{{x} _{1}+{x} _{2}}{2},\dfrac{{y} _{1}+{y} _{2}}{2}\right)$

$\therefore $ Modpoint of $OP=\left(\dfrac{0+0}{2},\dfrac{4+-4}{2}\right)$
$=\left(0,0\right)$

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 $y=-3$ 
           $a(10,-3)$