Tag: mid-point formula

Questions Related to mid-point formula

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

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

  1. 3

  2. $\displaystyle 3\sqrt{2}$

  3. $\displaystyle 2\sqrt{3}$

  4. $\displaystyle 2\sqrt{2}$

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Correct Option: B
Explanation:

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

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.

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

  1. (6, -5)

  2. (5, -6)

  3. (6, -6)

  4. (-6, 6)

Show answer
Correct Option: C
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,
$-1=\dfrac{y+4}{2}$
$=>-2=y+4$
$=>y=-6$

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

If (-2, -4) is the midpoint of (6, -7) and (x, y) then the values of x and y are

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

  1. x = 2, y = 1

  2. x = -10, y = -1

  3. x = 10, y = -1

  4. x = -8 , y = -1

  5. none of these

Show answer
Correct Option: C
Explanation:

Since, $(-2, -4)$ is the midpoint of $(6, -7)$ and $(x,y).$
$\Rightarrow \dfrac{x+6}2=-2\Rightarrow x=-10$
and $ \dfrac{y-7}2=-4\Rightarrow y=-1$
Option D is correct.

In the xy-plane, find the mid point of the line segment joining the points $\left( 5,9 \right) $ and $\left( 7,11 \right) $.

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

  1. $(1.5, 2)$

  2. $(6, 10)$

  3. $(5.5, 5)$

  4. $(6, -3.5)$

Show answer
Correct Option: B
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)$

The coordinates of points $P(-2, 2), Q(3, 2) $ and $R(3, -2)$ are the vertices of a rectangle $PQRS$`. What are the coordinates of S? 

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

  1. $(-3., -2)$

  2. $(-2, - 2)$

  3. $(3, 2)$

  4. $(2, 2)$

Show answer
Correct Option: B
Explanation:

Mid-point of $PR=\left(\cfrac{-2+3}2,\cfrac{2-2}2\right)=\left(\cfrac12,0\right)$

Let coordinate of $S=(x,y)$
Mid-point of $QS=\left(\cfrac{3+x}2,\cfrac{2+y}2\right)$
Mid-point of $PR=$Mid-point of $QS$
$\Rightarrow\left(\cfrac12,0\right)=$$\left(\cfrac{3+x}2,\cfrac{2+y}2\right)$
We have $\cfrac12=\cfrac{3+x}2\Rightarrow x=-2$ and $0=\cfrac{2+y}2\Rightarrow y=-2$
Coordinate of $S=(-2,-2)$
Hence, B is the correct option.

$M$ is the midpoint of the straight line $PQ$. If $P(-2,9)$ and $M$ is $(4,3)$, find the coordinates of $Q$.

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

  1. $(1,6)$

  2. $(10,-3)$

  3. $(10,6)$

  4. $(8,-3)$

Show answer
Correct Option: B
Explanation:
$\dfrac{x-2}{2}=4$ and $\dfrac{y+9}{2}=3$
 $\Rightarrow x=10$ and $y=-3$ 
           $a(10,-3)$

$M(2, 6)$ is the midpoint of $\overline {AB}$. If $A$ has coordinates $(10, 12)$, the coordinates of $B$ are

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

  1. $(6, 10)$

  2. $(-6, 0)$

  3. $(-8, -4)$

  4. $(18, 16)$

  5. $(22, 18)$

Show answer
Correct Option: B
Explanation:

Mid point $=$ $\dfrac{x _1+x _2}{2}, \dfrac{y _1+y _2}{2}$

Here mid-point $=(2,6)$
$x _1=10,y _1=12$
$\Rightarrow (2,6)=\dfrac{10+x _2}{2},\dfrac{12+y _2}{2}$
$\Rightarrow \dfrac{10+x _2}{2}=2; \dfrac{12+y _2}{2}=6$
$\Rightarrow x _2=4-10;y _2=12-12$
$\Rightarrow x _2=-6,y _2=0$
$\therefore $ co-ordinates of $B=(-6,0)$.

If the mid-point between the points $(a+ b, a- b)$ and $(-a, b)$ lies on the line $ax + by = k$, what is k equal to?

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

  1. $\dfrac ab$

  2. $a + b$

  3. $ab$

  4. $a - b$

Show answer
Correct Option: C
Explanation:

Mid point of points $(a+b,a-b),(-a,b)$ is $\left( \dfrac { a+b-a }{ 2 } ,\dfrac { a-b+b }{ 2 }  \right) =\left(\dfrac { b }{ 2 } ,\dfrac { a }{ 2 } \right)$

It lies on line $ax+by=k$ then 
$\Rightarrow a\times \dfrac{b}{2}+b\times \dfrac{a}{2}=k$
$\Rightarrow ab=k$

If a point $C$ be the mid-point of a line segment $AB$, then $AC = BC = (...) AB$.

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

  1. $3$

  2. $\dfrac{1}{2}$

  3. $2$

  4. $\dfrac{1}{4}$

Show answer
Correct Option: B
Explanation:

If $C$ is the midpoint of $AB$, then $C$ divides $AB$ in equal segments. Those segments are $AC$ and $AB$.

therefore, $AC=BC$. 
 $AC+BC=AB$ (As AC and BC are the segments of $AB$)
$\Rightarrow 2AC=2BC=AB\ \Rightarrow AC=BC=\dfrac{1}{2} AB$

Say true or false.
The mid-point of the line segment joining the points $P(x _1, y _1)$ and $Q(x _2, y _2)$ is 
$\dfrac {x _1+x _2}{2}, \dfrac {y _1+y _2}{2}.$
maths constructions mid-point formula midpoints division of a line segment

  1. True

  2. False

Show answer
Correct Option: A