Tag: mid-point of a line segment

Questions Related to mid-point of a line segment

Find the coordinates of the point which divides the line segment joining the points (6, 3) and (-4, 5) in the ratio 3 : 2 externally.
maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. (-12, 9)

  2. (-16, 9)

  3. (-24, 9)

  4. (-14, 9)

Show answer
Correct Option: C
Explanation:

Using the section formula, if a point $(x,y)$ divides the line joining the points

$({ x } _{ 1 },{ y } _{ 1 })$ and $({ x } _{ 2 },{ y } _{ 2 })$externally  in the ratio $ m:n $, then $(x,y) = \left(

\dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n } ,\dfrac { m{ y } _{ 2 }-n{ y } _{ 1 }

}{ m-n }  \right) $


Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (6,3) $ and

$({x } _{ 2 },{ y } _{ 2 }) = (-4,5) $  and $ m = 3, n = 2 $ in the section formula, we get 



$ C = \left( \dfrac { 3(-4)-2(6) }{ 3-2 } ,\dfrac { 3(5)-2(3) }{ 3-2 } 

\right) =\left( -24,9 \right) $

If the line joining A(2, 3) and B(-5, 7) is cut by x-axis at P then AP : PB is

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. 3 : 7

  2. -3 : 7

  3. 7 : 3

  4. 7 : -3

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Correct Option: B
Explanation:

Using the section formula, if a

point $(x,y)$ divides the line joining the points $({ x } _{ 1 },{ y } _{ 1

})$ and $({ x } _{ 2 },{ y } _{ 2 })$ in the ratio $ m:n $, then $(x,y) =

\left( \dfrac { m{ x } _{ 2 } + n{ x } _{ 1 } }{ m + n } ,\dfrac { m{ y } _{ 2

}  + n{ y } _{ 1 } }{ m + n }  \right) $


Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (2,3) $ and $({x } _{ 2 },{ y } _{ 2

}) = (-5,7) $  in the section formula, we get the point $ \left( \dfrac {

m(-5)  + n(2) }{ m + n } ,\dfrac { m(7) + n(3) }{ m + n }  \right)

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


As the point lies on x - axis, y -coordinate $ = 0 $.

$ => \dfrac { 7m + 3n }{ m + n} = 0 $ 

$ => 7m = -3n $  or $ m : n = -3:7 $

Find the coordinates of the point which divides the line segment joining the points $(6, 3)$ and $(-4, 5)$ in the ratio $3 : 2$, externally.

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. $(24,9)$

  2. $(-24,-9)$

  3. $(-24,9)$

  4. $(24,-9)$

Show answer
Correct Option: C
Explanation:

Let P$(x,y)$ be the required point.
Using the section formula, if a point $(x,y)$ divides the line joining the points $({ x } _{ 1 },{ y } _{ 1 })$ and $({ x } _{ 2 },{ y } _{ 2 })$externally  in the ratio $ m:n $, then $(x,y) = \left( \dfrac { m{ x } _{ 2 }-n{ x } _{ 1 } }{ m-n } ,\dfrac { m{ y } _{ 2 }-n{ y } _{ 1 } }{ m-n }  \right) $
Substituting $({ x } _{ 1 },{ y } _{ 1 }) = (6,3) $ and $({x } _{ 2 },{ y } _{ 2 }) = (-4,5) $  and $ m = 3, n = 2 $ in the section formula, we get 

$ P = \left( \dfrac { 3(-4)-2(6) }{ 3-2 } ,\dfrac { 3(5)-2(3) }{ 3-2 } \right) =\left(-24,9 \right) $

The ordinate of the point which divides the line joining the origin and the point (1, 2) externally in the ratio of 3 : 2 is

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. $-2$

  2. $\displaystyle\frac{3}{5}$

  3. $\displaystyle\frac{2}{5}$

  4. $6$

Show answer
Correct Option: D
Explanation:

The co-ordinates of the required point will be

$\displaystyle y=\dfrac{m _1y _2-m _2y _1}{m _1-m _2}$

$\displaystyle=\dfrac{3\times2-2\times0}{3-2}=6$

Find the co-ordinates of a point C on AB produced such that $3AB = AC$, where $A = (3, 2)$ and $B = (-2, 4).$

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. $(-12, 8)$

  2. $(8, 12)$

  3. $(12, 8)$

  4. $(-8, 12)$

Show answer
Correct Option: A
Explanation:

From the above condition, we can observe that the point B  divides the line segment joining AC in 1;3 ratio, or $AC:AB=3:1$ or $AB:BC=2:1$. Let the coordinates of C be (x,y). Therefore,
$B(-2,4)=\left(\dfrac{1(x)+2(3)}{3},\dfrac{1(y)+2(2)}{3}\right)$
Or  $\dfrac{6+x}{3}=-2$ or $6+x=-6$ or $x=-12$. Similarly $\dfrac{4+y}{3}=4$ or $4+y=12$ or $y=8$.
Hence $C(x,y)=(-12,8)$

Find $x$ and $y$ if $(2,5)$ is the midpoint of points $(x,y)$ and $(-5,6)$.

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. $x=4, y=9$

  2. $x=9, y=4$

  3. $x=-9, y=4$

  4. $x=9, y=-4$

Show answer
Correct Option: B
Explanation:
If the end points of a line segment is $(x,y)$ and $(-5,6)$ then the midpoint of the line segment has the coordinates:
$\left( \dfrac { x-5 }{ 2 } ,\dfrac { y+6 }{ 2 }  \right) =\left( 2,5 \right)$ ...(hint: using mid-point formula)
Now equating the points:  $\dfrac { x-5 }{ 2 } =2$ 
$\Rightarrow x-5=4$ 
$\Rightarrow x=9$
And, $\dfrac { y+6 }{ 2 } =5$ 
$\Rightarrow y+6=10$ 
$\Rightarrow y=4$

Hence, $x=9$ and $y=4$.

Find the coordinates of the point which divides the join of the points $(2,4)$ and $(6,8)$ externally in the ratio $5:3$.

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. $(12,14)$

  2. $(14,12)$

  3. $(-12,14)$

  4. $(12,-14)$

Show answer
Correct Option: A
Explanation:

Given $A(2, 4)$ and $B(6,8)$

Applying the section formula externally,

$\left( \dfrac { L{ x } _{ 2 }-{ mx } _{ 1 } }{ L-m } ,\dfrac { L{ y } _{ 2 }-{ my } _{ 1 } }{ L-m }  \right)$

Here the ratio given is $5:3$ that is $L=5$ and $m=3$, therefore,

$\left( \dfrac { L{ x } _{ 2 }-{ mx } _{ 1 } }{ L-m } ,\dfrac { L{ y } _{ 2 }-{ my } _{ 1 } }{ L-m }  \right) =\left( \dfrac { (5\times 6)-(3\times 2) }{ 5-3 } ,\dfrac { (5\times 8)-(3\times 4) }{ 5-3 }  \right) $


$=\left( \dfrac { 30-6 }{ 2 } ,\dfrac { 40-12 }{ 2 }  \right) =\left( \dfrac { 24 }{ 2 } ,\dfrac { 28 }{ 2 }  \right) =\left( 12,14 \right)$ 

Hence, the coordinates of the point is $(12,14)$.

If the join of the two points $(x _1, y _1)$, $(x _2, y _2)$ is divided by a point R externally in ratio $m : n$ then

maths transformations midpoint of line segment mid point formula mid-point of a line segment

  1. x - coordinates is $\dfrac {mx _2 - nx _1}{m - n}$

  2. x - coordinates is $\dfrac {my _2 - ny _1}{m - n}$

  3. Both (a) and (b) above

  4. None of these

Show answer
Correct Option: A
Explanation:

When a point C divides a segment $A(x _1,y _1)$ and $B(x _2,y _2)$ in the ratio $m:n$ externally, we use the section formula to find the coordinates of that point.

The Coordinates of point R will be,

$X=\dfrac{mx _2-nx _1}{m-n}$.

If $z = \cos \dfrac{\pi }{6} + i\sin \dfrac{\pi }{6}$, then

maths introduction to three dimensional geometry midpoint of line segment mid point formula mid-point of a line segment

  1. $\left| z \right| = 1,\arg z = \dfrac{\pi }{4}$

  2. $\left| z \right| = 1,\arg z = \dfrac{\pi }{6}$

  3. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = \dfrac{{5\pi }}{{24}}$

  4. $\left| z \right| = \dfrac{{\sqrt 3 }}{2},\arg z = {\tan ^{ - 1}}\dfrac{1}{{\sqrt 2 }}$

Show answer
Correct Option: B
Explanation:

if $z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)$ then.

As $z=|z|e^{i arq (z)}$
$\therefore z=\cos \left(\dfrac{\pi}{6}\right)+i\sin \left(\dfrac{\pi}{6}\right)=e^{i\left(\pi/6\right)}\quad [\because e^{i\theta}=\cos\theta+i\sin \theta]$
$\Rightarrow |z|=1,  arq=\dfrac{\pi}{6}$

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

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

  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

Show answer
Correct Option: D
Explanation:

Statement -1 is false,

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