Question

Prove that the lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one another.

Answer 1

Hint: - Here, we made a quadrilateral whose one diagonal is x axis. And then suppose the coordinate of the corner of the quadrilateral with respect to x axis and y axis. And then go through the bisector formula of coordinates.

Let ABCD be the quadrilateral such that diagonal AC is along x axis suppose the coordinates A, B, C and D be $$(0,0),(x_2,y_2)(x_1,0)$$and $$(x_3,y_3)$$ respectively.
E and F are the mid points of sides AD and BC respectively, G and H are the midpoint of diagonals AC and BD. And the point of intersection of EF and GH is I.
Coordinates of E are $$\left({\displaystyle \frac{0+x_3}{2}},{\displaystyle \frac{0+y_3}{2}}\right)=\left({\displaystyle \frac{x_3}{2}},{\displaystyle \frac{y_3}{2}}\right)$$
Coordinates of Fare$$\left({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{0+y_2}{2}}\right)=\left({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_2}{2}}\right)$$
Coordinates of midpoint of EF are $$\left({\displaystyle \frac{{\displaystyle \frac{x_3}{2}}+{\displaystyle \frac{x_1+x_2}{2}}}{2}},{\displaystyle \frac{{\displaystyle \frac{y_3}{2}}+{\displaystyle \frac{y_2}{2}}}{2}}\right)=\left({\displaystyle \frac{x_1+x_2+x_3}{4}},{\displaystyle \frac{y_2+y_3}{4}}\right)$$
G and H are the mid points of diagonal AC and BD respectively then

Coordinates of G are$$\left({\displaystyle \frac{0+x_1}{2}},{\displaystyle \frac{0+0}{2}}\right)=\left({\displaystyle \frac{x_1}{2}},0\right)$$
Coordinates of H are$$\left({\displaystyle \frac{x_2+x_3}{2}},{\displaystyle \frac{y_2+y_3}{2}}\right)$$
Coordinates of midpoint of GH are $$\left({\displaystyle \frac{{\displaystyle \frac{x_1}{2}}+{\displaystyle \frac{x_2+x_3}{2}}}{2}},{\displaystyle \frac{{\displaystyle \frac{y_2}{2}}+{\displaystyle \frac{y_2+y_3}{2}}}{2}}\right)=\left({\displaystyle \frac{x_1+x_2+x_3}{4}},{\displaystyle \frac{y_2+y_3}{4}}\right)$$
As you can see, midpoints of both EF and GH are the same. So, EF and GH meet and bisect each other.
Hence, proved.

Note:-Whenever we face such types of questions first of all make the diagram by the statement given by the question, then use the section formula to find the midpoint of a line. If the two points have the same value then it must coincide.