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

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

R is the midpoint of the segment $\bar{PT}$, and $Q$ is the midpoint of line segment $\bar{PR}$. If $S$ is a point between $R$ and $T$ such that the length of segment $\overline{QS}$ is $10$ and the length of segment $\overline{PS}$ is $19$, what is the length of segment $\overline{ST}$?

  1. $13$
  2. $14$
  3. $15$
  4. $16$
  5. $17$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Given that $PR=RT$ and $PQ=QR$

Let $QR=PQ=x$ , we get $PR=RT=2x$
Given that $S$ is a point between $R$ and $T$
Given $QS=10$ , $PS=19$
$PS=PQ+QS=PQ+10=19$
$\Rightarrow PQ=9$
Therefore we get $PT=4x=36$
$\Rightarrow ST=PT-PS=36-19=17$

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

$P and Q$ are points on the line joining $A(-2,5) and (3,1)$ such that $AP=PQ=QB$ then the mid point of $PQ$ is ?

  1. $\dfrac{1}{23}$
  2. $\dfrac{-1}{24}$
  3. $\dfrac{-24}{1}$
  4. $(1,4)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since $AP = PQ = BQ$


hence mid point of PQ will be the mid point of AB.



Thus mid point of AB = [( –2+3)/2 , (5+1)/2 ]





mid point of AB = [1/2 , 3] = mid point of PQ



 

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 maths parts and whole finding the whole when a fraction is given multiplication of a fraction multiplication of a fractions

Which point on the number line most likely represent $-2\cfrac { 5 }{ 8 } $?

  1. On the left of $-3$
  2. On the right of $-2$
  3. Between $-2$ and $-3$
  4. In the middle of $-2$ and $-3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$-2\cfrac { 5 }{ 8 } $

$-2\cfrac { 5 }{ 8 } =(-1)2\cfrac { 5 }{ 8 } $

$=-1\times \cfrac { 21 }{ 8 } $

$=-2.625$

It lies in between $-3$ and $-2$
Multiple choice maths the plane equation of a plane in intercept form equation of a plane in different forms different forms of equation of a plane

The sum of the intercepts of the plane which bisects the line segment joining $(0,1,2)$ and $(2,3,0)$ perpendicularly is

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

The dr's of the normal to the plane will be $(2,2,-2)$.
Hence, the plane is , $2x + 2y - 2z = d$
The midpoint of the line segment is $(1,2,1)$.
Hence, $d = 4$.
Hence, the intercepts are $\left (  \dfrac{1}{2} , 1 ,  \dfrac{1}{2} \right )$
The sum of intercepts is $2$.

Multiple choice maths geometry similarity and right angled triangle angle theorems for a right angled triangle apollonius's theorem

P, Q, R are the points of intersection of a line 1 with sides BC, CA, AB of a $\Delta$ ABC 
respectively, then $\dfrac{BP}{PC} \dfrac{CQ}{QA} \dfrac{AR}{RB}$

  1. 1

  2. 2

  3. -1

  4. -2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a direct application of Menelaus' Theorem, which states that for a line intersecting the sides of a triangle, the product of the ratios of the segments is 1.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

The line joining the mid points of the diagonals of a trapezium has length $3$cm. If the longer base is $97$cm then the shorter base is:

  1. $94$cm
  2. $92$cm
  3. $91$cm
  4. $90$cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The line joining the mid point of the diagonals of a trapezium is half the length of the difference between the two sides.
Let the smaller side be $x$
Then, $3 = \dfrac{97 -x}{2}$
$6= 97 - x$
$x = 91$ cm