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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.
Solution
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)`
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