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Mid Point Formula

Coordinate Geometry I - Concepts
Distance Formula
Section Formula
Mid Point Formula
Trisection Points of Line
Class - 10th CBSE Subjects
 
 
Concept Explanation
 

Mid-Point Formula

Coordinates of the mid-point M (x,y) of the segment AB are obtained by taking m_1 = m_2 in the section formula:

x=frac{m_{2}x_{1}+m_{1}x_{2}}{m_{1}+m_{2}};and;y=frac{m_{2}y_{1}+m_{1}y_{2}}{m_{1}+m_{2}}

x=frac{m_{1}x_{1}+m_{1}x_{2}}{m_{1}+m_{1}};and;y=frac{m_{1}y_{1}+m_{1}y_{2}}{m_{1}+m_{1}}

 

x=frac{m_{1}(x_{1}+x_{2})}{2m_{1}};and;y=frac{m_{1}(y_{1}+y_{2})}{2m_{1}}   

     large x=frac{1}{2}(x_{1}+x_{2}),          large y=frac{1}{2}(y_{1}+y_{2}),

Illustration: Find the distance of the point (1,2) from the mid-point of the line segment joining the points (6,8) and (2,4).

Solution:   Let the points (1,2), (6,8), and (2,4) be denoted by A, B, and C respectively. Let M be the mid-point of BC. Coordinates of M, the mid-point of BC, are given by

           large left ( frac{6+2}{2},frac{8+4}{2} right )=(4,6)

We have          large AM^{2}=(4-1)^{2}+(6-2)^{2}

                                      large =3^{2}+4^{2}=9+16=25

large Rightarrow                large AM=sqrt{25}=5;units

Illustration: Find the lengths of medians of the triangle with vertices A(2,2), B(0,2), and C(2,-4).

Solution:   Let the coordinates of A, B, and C be (2, 2), (0,2), and (2, -4) respectively. Let D, E, F be the mid-points of BC, CA, and AB respectively.

Then the coordinates of D, E, F are given by

            large Dleft ( frac{0+2}{2},frac{2-4}{2} right ), large large Eleft ( frac{2+2}{2},frac{2-4}{2} right ), large Fleft ( frac{0+2}{2},frac{2+2}{2} right ) 

or D (1, -1) E ( 2, -1) , F (1, 2)

large therefore   Length of median AD=  large sqrt{(1-2)^{2}+(-1-2)^{2}}=sqrt{10}

      Length of median BE= large sqrt{(2-0)^{2}+(-1-2)^{2}}=sqrt{13}

      Length of median CF = large sqrt{(1-2)^{2}+[2-(-4)]^{2}}=sqrt{37}

large therefore Lengths of medians are large sqrt{10},sqrt{13},sqrt{37};;units.

Illustration: Three consecutive vertices of a parallelogram are A(1,2), B(1,0) C(4,0). Find the fourth vertex D.

Solution:  Let coordinates of D be (x,y). As ABCD is a parallelogram, the diagonals AC and BD bisect each other. If M is the mid-point of AC, then coordinates of M are large left ( frac{1+4}{2},frac{2+0}{2} right )=left ( frac{5}{2},1 right )

As M is the mid-point of BD also, coordinates of M are large left ( frac{1+x}{2},frac{0+y}{2} right )=left ( frac{1+x}{2},frac{y}{2} right ).

We have   large left ( frac{1+x}{2},frac{y}{2} right )=left ( frac{5}{2},1 right ) Rightarrow frac{1+x}{2}= frac{5}{2};and;frac{y}{2}=1

large Rightarrow 1+x=5;and;y=2Rightarrow x=4,y=2

large therefore    Coordinates of D are (4,2)

Coordinates of Centroid of Triangle

The centroid of a triangle is the point of concurrence of the medians of a triangle.

If G is the centroid of triangle ABC, then G divides the median AD in the ratio 2:1.

Let D be the mid-point of BC, the coordinates of D are large left ( frac{x_{2}+x_{3}}{2},frac{y_{2}+y_{3}}{2} right ).

As G(x,y) divides AD in the ratio 2 : 1

   large x=frac{1(x_{1})+2left ( frac{x_{2}+x_{3}}{2} right )}{2+1}=frac{x_{1}+x_{2}+x_{3}}{3}

and  large y=frac{1(y_{1})+2left ( frac{y_{2}+y_{3}}{2} right )}{2+1}=frac{y_{1}+y_{2}+y_{3}}{3}

Thus, the coordinates of the centroid are

large left ( frac{x_{1}+x_{2}+x_{3}}{3},frac{y_{1}+y_{2}+y_{3}}{3} right )

Illustration: Find the third vertex of the triangle whose two vertices are (-3,1) and (0,-2) and the centroid is the origin.

Solution:  Let the two given vertices be A(-3,1) and B(0,-2). Let the third vertex be C(x,y) . Coordinates of the centroid of large Delta ABC is given by

 large left ( frac{-3+0+x}{3},frac{1-2+y}{3} right )=left ( frac{-3+x}{3},frac{-1+y}{3} right ).

As the centroid of large Delta ABC is given to be the origin,

we have large left ( frac{-3+x}{3},frac{-1+y}{3} right )=(0,0)

large Rightarrow ;;-3+x=0,;;-1+y=0

large Rightarrow ;;x=3,;;y=1

Thus, the coordinates of the third vertex are (3,1).

Sample Questions
(More Questions for each concept available in Login)
Question : 1

If   p=(frac{a}{3},4)  is the mid point of the segment joining the points  Q (-6,5) and R (-2,3) .  Then the value of a  is : 

Right Option : D
View Explanation
Explanation
Question : 2

If the mid-point of the line segment joining the points A(3,4) and B(k,6) is P(x,y) and x+y-10=0,Find the value of k.

Right Option : A
View Explanation
Explanation
Question : 3

C is the mid-point of PQ, if P is (4,x), C is (y,-1) and Q is (-2,4), then x and y respectively are

Right Option : A
View Explanation
Explanation
 
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