Question

A line segment AB is increased along its length by 25% by producing it to C on the side of B. If A and B have the coordinates (−2, −3) and (2, 1) respectively, then find the coordinates of C.

Answer 1

Let the point C be (a, b)

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

AB = `sqrt((2 + 2)^2 + (1 + 3)^2`

= `sqrt(4^2 + 4^2)`

= `sqrt(16 + 16)`

= `4sqrt(2)`

BC = 25% of AB

= `25/100 xx 4sqrt(2)`

= `sqrt(2)`

`"AB"/"BC" = (4sqrt(2))/sqrt(2)` = 4

The ratio is 4 : 1 (m : n)

A line divides internally in the ratio m : n

The point P = `(("m"x_2 + "n"x_1)/("m" + "n"), ("m"y_2 + "n"y_1)/("m" + "n"))`

The point B = `(4"a" - 2)/(4 + 1), (4"b" - 3)/(4 + 1)`

(2, 1) = `(4"a" - 2)/5, (4"b" - 3)/5`

`(4"a" - 2)/5` = 2

4a – 2 = 10

4a = 12

a = `12/4` = 3

and

`(4"b" - 3)/5` = 1

4b – 3 = 5

4b = 8

b = `8/4` = 2

The co-ordinate of C is (3, 2)