Question

Find the lengths of the medians of a triangle whose vertices are A (−1, 3), B (1, −1), and C (5, 1).

Answer 1

We have to find the lengths of the medians of a triangle whose co-ordinates of the vertices are A (−1, 3); B (1, −1), and C (5, 1).

So we should find the midpoints of the triangle’s sides.

In general to find the mid-point P (x, y) of two points A (x₁, y₁) and B (x₂, y₂) we use section formula as,

`P (x, y) = ((x_1 + x_2) / 2, (y_1 + y_2) / 2)`

Therefore, mid-point P of side AB can be written as,

`P (x, y) = ((-1 + 1) / 2, (3 -1) / 2)`

Now equate the individual terms to get,

x = 0

y = 1

So co-ordinates of P are (0, 1)

Similarly, mid-point Q of side BC can be written as,

`Q (x, y) = ((5 + 1)/2, (1 - 1) / 2)`

Now equate the individual terms to get,

x = 3

y = 0

So co-ordinates of Q are (3, 0)

Similarly, mid-point R of side AC can be written as,

`R (x, y) = ((5 - 1) / 2, (1 + 3) / 2)`

Now equate the individual terms to get,

x = 2

y = 2

So co-ordinates of Q are (2, 2)

Therefore length of the median from A to the side BC is,

AQ = `sqrt((-1 -3)^2 + (3 - 0)^2)`

= `sqrt(16 + 9)`

= 5

Similarly length of the median from B to the side AC is,

BR = `sqrt((1- 2)^2 + (-1 - 2)^2)`

= `sqrt(1 + 9)`

= `sqrt(10)`

Similarly length of the median from C to the side AB is

CP = `sqrt((5 - 0)^2 + (1 - 1)^2)`

= `sqrt25`

= 5