Question

In the given figure, point D divides the side BC of ΔABC in the ratio 1 : 2. Find the length AD.

Answer 1

Given coordinates: B(–2, 1), C(4, 2) and ratio m : n = 1 : 2.

Let D have coordinates (x, y).

`D(x, y) = ((mx_2 + nx_1)/(m + n), (my_2 + ny_1)/(m + n))`

`x = (1(4) + 2(-2))/(1 + 2)`

= `(4 - 4)/3`

= 0

`y = (1(2) + 2(1))/(1 + 2)`

= `(2 + 2)/3`

= `4/3`

So, the coordinates of D are `(0, 4/3)`.

Now, we find the length AD where A is (1, 5):

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

`AD = sqrt((0 - 1)^2 + (4/3 - 5))`

`AD = sqrt((-1)^2 + ((4 - 15)/3)^2`

`AD = sqrt(1 + ((-11)/3)^2`

`AD = sqrt(1 + 121/9)`

`AD = sqrt((9 + 121)/9)`

`AD = sqrt(130/9)`

`AD = sqrt(130)/3` units

The length of AD is `sqrt(130)/3` units.