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

The mid point of $(2,3)$ and $(8,9)$ is 

  1. $(5,6)$
  2. $(2,8)$
  3. $(5,7)$
  4. $(4,6)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The points are $(x _1,y _1)=(2,3)$ and $(x _2,y _2)=(8,9)$


The mid point is given as $\left(\dfrac{2+8}2,\dfrac {3+9}2\right)$
                                
$\left(\dfrac {10}2,\dfrac {12}2\right)=(5,6)$

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

The mid point of $(3,4)$ and $(1,-2)$

  1. (2,1)

  2. (1,2)

  3. (2,-1)

  4. (1,-2)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The points are $(3,4)$ and $(1,-2)$


The mid point of $(3,4)$ and $(1,-2)$ is given by 

$\left(\dfrac {x _1+x _2}2,\dfrac {y _1+y _2}2\right)\\\left(\dfrac{3+1}2,\dfrac {4-2}2\right)=(2,1)$

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

A triangle has vertices A(1,-1) B(2,4) and C(6,0) The length of the median from A is

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

Midpoint of BC = L = (4, 2)
$\displaystyle \therefore AL=\sqrt{\left ( 1-4 \right )^{2}+\left ( -1-2 \right )^{2}}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}$

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

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