Question

If A (–1, 3), B (1, –1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

Answer 1

The distance d between two points `(x_1,y_1)` and `(x_2, y_2)` is given by the formula

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

The co-ordinates of the midpoint `(x_m, y_m)` between two points `(x_1, y_1)` and `(x_2, y_2)` is given by,

`(x_m,y_m) = (((x_1 + x_2)/2)"," ((y_1 + y_2)/2))`

Here, it is given that the three vertices of a triangle are A(−1, 3), B(1, −1) and C(5, 1).

The median of a triangle is the line joining a vertex of a triangle to the mid-point of the side opposite this vertex.

Let ‘D’ be the mid-point of the side ‘BC’.

Let us now find its co-ordinates.

`(x_D,y_D) = (((1 + 5)/2)"," ((-1+1)/2))`

`(x_D, y_D) = (3,0)`

Thus we have the co-ordinates of the point as D(3, 0).

Now, let us find the length of the median ‘AD’.

`AD = sqrt((-1-3)^2 + (3 -  0)^2)`

`= sqrt((-4)^2 + (3)^2)`

`= sqrt(16 + 9)`

AD = 5

Thus the length of the median through the vertex ‘A’ of the given triangle is 5 units.