Question

Find the equations of the median through vertex A of the triangle whose vertices are A(2, 5), B(−4, 9), and C(−2, −1).

Answer 1

Here, vertex A has the opposite side, which is BC,

Let’s find the midpoint (D) of side BC,

The coordinates of the midpoint (x, y) of a line segment joining (x₁, y₁) and (x₂, y₂) are:

`M = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

For B(−4, 9), and C(−2, −1):

`D = ((-4 + (-2))/2, (9 + (-1))/2)`

`D = (-6)/2, 8/2`

∴ D = (−3, 4)

Now, find the slope of the line passing through A(2, 5), D(−3, 4),

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (4 - 5)/(-3 - 2)`

`m = (-1)/(-5)`

∴ `m = 1/5`

Using the point–slope formula:

y − y₁ = m(x − x₁)

Substituting point A(2, 5), and slope m = `1/5`:

`y - 5 = 1/5 (x - 2)`

Let’s write the above equation in standard form (Ax + By + C = 0),

5(y − 5) = 1(x − 2)      ...[Multiplied both sides by 5 to eliminate the fraction.]

5y − 25 = x − 2

x − 5y − 2 + 25 = 0

∴ x − 5y + 23 = 0

Hence, the equation of the median through vertex A is x − 5y + 23 = 0.