Question

Let A be `(−3,5)` and B be `(5,−10))`. Find: (1) the length of segment `bar(AB)` (2) the midpoint `P` of `bar(AB)` (3) the point `Q` which splits `bar(AB)` in the ratio `2:5`?

Answer 1

(1) the length of the segment `bar(AB)` is `17`
(2) Midpoint of `bar(AB)` is `(1,-7 1/2)`
(3) The coordinates of the point `Q` which splits `bar(AB)` in the ratio `2:5` are `(-5/7,5/7)`

Explanation:

If we have two points `A(x_1,y_1)` and `B(x_2,y_2)`, length of `bar(AB)` i.e. distance between them is given by

`sqrt((x_2-x_1)^2+(x_2-x_1)^2)`

and coordinates of the point `P` that divides the segment `bar(AB)` joining these two points in the ratio `l:m` are

`((lx_2+mx_1)/(l+m),(lx_2+mx_1)/(l+m))`

and as midpoint divided segment in ratio `1:1`, its coordinated would be `((x_2+x_1)/2,(x_2+x_1)/2)`

As we have `A(-3,5)` and `B(5,-10)`

(1) the length of the segment `bar(AB)` is

`sqrt((5-(-3))^2+((-10)-5)^2)`

= `sqrt(8^2+(-15)^2)=sqrt(65+225)=sqrt289=17`

(2) Midpoint of `bar(AB)` is `((5-3)/2,(-10-5)/2)` or `(1,-7 1/2)`

(3) The coordinates of the point `Q` which splits `bar(AB)` in the ratio `2:5` are

`((2xx5+5xx(-3))/7,(2xx(-10)+5xx5)/7)` or `((10-15)/7,(-20+25)/7)`

i.e. `(-5/7,5/7)`