Question

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

Answer 1

LetΔOAB be any triangle such that O is the origin and the other co-ordinates areA(x₁y₁):(x₂,y₂). P and R are the mid-points of the sides OA and OB respectively.

We have to prove that line joining the mid-point of any two sides of a triangle is equal to half of the third side which means, 

`PR=1/2(AB)` 

In general to find the mid-pointp(x,y) of two points `A(x_1,x_2)`and `B(x_2,y_2)` we use section formula as, 

`P(x,y)=((x_1+x_2)/2,(y_1+y_2) /2)` 

so, 

Co-ordinates of p is, 

`P(x_1/2,y_1/2)` 

Similarly, co-ordinates of R is, 

`R(x_2/2,y_2/2)` 

In general, the distance between A`(x_1,y_1)` and B`(x_2,y_2)`is given by,  

`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

Similarly, 

`PR=sqrt((x_2/2-x_1/2)^2+(y_2/2-y_1/2)^2)` 

`=1/2 sqrt((x_2-x_1)2+(y_2-y_1)2 ` 

`1/2(AB) ` 

Hence, 

`PR=1/2(AB)`