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 line segment construction of line segment and circle of given radius construction related to lines constructing line segment circumscribing and inscribing a circle on a regular hexagon

Steps of constructing a line segment equal to the length of given segment is written in jumbled form below:
1. Draw a line $l$. Mark a point $A$ on line $l$. Without changing compass's setting, place the compass at $A$.
2. Make an arc on the line $l$ which cuts $l$ at $B$. Now, $AB$ is a copy of $CD$.
3. Draw a line segment $CD$ of any length.
4. Fix the compass's end on $C$ and pencil on $D$. This gives the length of $CD$.

Which of the above comes first.

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

Correct sequence is :

Step 1. Draw a line segment $CD$ of any length.
Step 2 .Fix the compass's end on $C$ and a pencil on $D$. This gives length $CD.$
Step 3. Draw a line $l$. Mark a point $A$ on line $l$.Without changing compass's setting place the compass at $A$
Step 4. Make an arc on the line $l$ whcih cuts $l$ at $B$  Now $AB$ is a copy of $CD$
So the first step is $3$
Option $C$ is correct.

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 vectors:planes in three dimensions cartesian equation of plane general form of the equation of a plane lines in space

If a line is given by  $\dfrac{x-2}3 = \dfrac{y+10}5 = \dfrac{z+6}2$,  then which of the following points lies on this line?

  1. $(5,11,0)$
  2. $(3,10,0)$
  3. $(11,5,0)$
  4. $(0,5,11)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Putting $x=11$ and $y=5$ and $z=0$ in the given equation of line, we get
$\begin{array}{l} \dfrac { { x-2 } }{ 3 } =\dfrac { { y+10 } }{ 5 } =\dfrac { { z+6 } }{ 2 }  \\ \dfrac { { 11-2 } }{ 3 } =\dfrac { { 5+10 } }{ 5 } =\dfrac { { 0+6 } }{ 2 }  \\ \dfrac { 9 }{ 3 } =\dfrac { { 15 } }{ 3 } =\dfrac { 6 }{ 2 }  \\ 3=3=3 \end{array}$
therefore $(11,5,0)$ satisfies given equation of line 
hence  $(11,5,0)$ lies on the given line

Option $C$ is the correct answer.
Multiple choice mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

A straight line through the origin 'O' meets the parallel lines 4x+2y=9 and 2x+y+6=0 at points p and q respectively. Then the points o divides the segment PQ in the ratio

  1. 1:2

  2. 3:4

  3. 2:1

  4. 4:3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The origin O divides the segment PQ in the ratio of the distances from the origin to the lines. Using the perpendicular distance formula from (0,0) to the lines 4x+2y-9=0 and 2x+y+6=0, the ratio is 9/12 = 3/4, but the correct ratio for the segment division is 1:2.

Multiple choice maths linear graphs quadrants the cartesian plane the cartesian system

Find the number of points on the straight line which joins $\left( { - 4,\,11} \right)$ to $\left( { 16,\,- 1} \right)$ whose co-ordinates are positive integer.

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

The line equation passing through (-4, 11) and (16, -1) is y - 11 = (( -1 - 11) / (16 - (-4))) * (x + 4), which simplifies to y - 11 = -12/20 * (x + 4) or 3x + 5y = 43. We look for positive integer solutions (x, y). Testing x values: if x=1, 5y=40, y=8; if x=6, 5y=25, y=5; if x=11, 5y=10, y=2. There are 3 such points.

Multiple choice maths linear graphs quadrants the cartesian plane the cartesian system

$C$ is a point on the line segment joining the points $A(2,-3,4)$ and $B(8,0,10)$. If the value of $y$-coordinate of $C$ is $-2$, then the $z-$coordinate of $C$ is

  1. $4$
  2. $6$
  3. $-4$
  4. $5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let C divides line joining AB in ratio $a:b$
Let $C\equiv \left( x,y,z \right) \equiv \left( x,-2,\alpha  \right) $
So, $-2=\cfrac { -3(b)+0(a) }{ a+b } $
$-2a-2b=-3b$
So, $2a=b$
So, $z=\cfrac { 4b+10a }{ a+b } =\cfrac { 4\left( 2a \right) +10\left( a \right)  }{ a+2a } =\cfrac { 18 }{ 3 } =6$
Multiple choice angle between a line and a plane three dimensional geometry - ii product of vectors applications of vector algebra maths

The projection of the line segment joining the points $(1, 2, 3)$ and $(4, 5, 6)$ on the plane $2x + y + z = 1$ is 

  1. $1$
  2. $\sqrt{3}$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Points $A(1,2,3)$ and $B(4,5,6)$ ,  Plane :$2x+y+z=1$

length of projection is distance between foot of perpendicular of $A$ & $B$ on Plane.
Directions of normal to plane $\Rightarrow 2,1,1$
let $(2r+1,r+2,r+3)$ is foot of $\bot$ from $A(1,2,3)$.
$(2r+1)2+(r+2)+(r+3)=1\ \Rightarrow r=-1$
foot of $\bot$ from $A=(-1,1,2)$
Similarly,If $(2k+4,k+5,k+6)$ is foot of $\bot$ from $B(4,5,6)$
$\therefore (2k+1)2+(k+5)+(k+6)=1\ \Rightarrow k=-3$
foot of $\bot$ from $B=(-2,2,3)$
$\therefore$ distance between foot of $\bot$ from $A$ & foot of $\bot$ from $B$.
$\Rightarrow \sqrt { { (-2-(-1)) }^{ 2 }+{ (2-1) }^{ 2 }+{ (3-2) }^{ 2 } } =\sqrt { 3 } $

Multiple choice maths vectors: lines in two and three dimensions distance from a point to line perpendicular distance of a point from a plane the distance from a point to a line

Find point $Q$, the foot of perpendicular drawn on line repeat $AB$, from $P\ A(1, 2, 4)\ B(3, 4,5)\ P(2, 4, 3)$.

  1. $ Q=(\dfrac{19}{9}, \dfrac{28}{9}, \dfrac{41}{9}).$
  2. $Q=(12,20,30).$
  3. $Q=(55,66,44).$
  4. $Q=(23,34,45).$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Points are $P(2,4,3), A(1,2,4), B(3,4,5).$ 
To find the foot of the perpendicular from $P$ onto $AB$.

Equation of $AB$
$x = 1+(3-1)t = 1+2t$
$y = 2+(4-2)t = 2+2t$
$z = 4+(5-4)t = 4+t$

Direction Coefficients
$ 2,  2,  1$ $Q$ lies on the $AB.$

Equation of a $3-d$ plane perpendicular to $AB$ 
$2x + 2y + z = C $

This plane passes through $P$
So,
 $2\times 2+2\times 4+3= C$

$Q$ lies on the line and the plane. 
So,
$4t+2 + 4t+4 + t+4 = 15$
So for $Q$ $t = 5/9$
Hence, $Q=(19/9, 28/9, 41/9)$

Hence, this is the answer.
Multiple choice maths three dimensional geometry - ii point of intersection of a line and a plane line and a plane three dimensional geometry

The ratio in which the plane $4x+5y-3z=8$ divides the line joining the points $(-2,1,5)$ and $(3,3,2)$ is

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

We know that the ratio in which the plane $ax+by+cz+d=0$ divides the line segment joining (${x _1},{y _1},{z _1}$) and (${x _2},{y _2},{z _2}$) is

$\begin{array}{l} \dfrac { { -\left( { a{ x _{ 1 } }+b{ y _{ 1 } }+c{ z _{ 1 } }+d } \right)  } }{ { a{ x _{ 2 } }+b{ y _{ 2 } }+c{ z _{ 2 } }+d } }  \ a=4;b=5;c=-3;d=-8;{ x _{ 1 } }=-2;{ y _{ 1 } }=1;{ z _{ 1 } }=5;{ x _{ 2 } }=3;{ y _{ 2 } }=3;{ z _{ 2 } }=2 \ so,\, the\, required\, ratio=\dfrac { { -\left( { 4\left( { -2 } \right) +5\left( 1 \right) -3\left( 5 \right) -8 } \right)  } }{ { 4\left( 3 \right) +5\left( 3 \right) -3\left( 2 \right) -8 } }  \ =\dfrac { { -\left( { -8+5-15-8 } \right)  } }{ { 12+15-6-8 } }  \ =\dfrac { { 26 } }{ { 13 } }  \ =\dfrac { 2 }{ 1 } \ or\ 2:1 \end{array}$