Question

А(2, −4), В(3, 3) and C(−1, 5) are the vertices of triangle ABC. Find the equation of the altitude of the triangle through B.

Answer 1

The altitude through B is perpendicular to the opposite side BC.

Using the slope formula with A(2, −4) and C(−1, 5):

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

`m_(AC) = (5 - (-4))/(-1 - 2)`

`m_(AC) = 9/-3`

∴ m_AC = −3

Since the altitude is perpendicular to AC, m_alt × (−3) = −1:

`m_"alt" = 1/3`

Using the point-slope formula with the altitude through B(3, 3):

y − y₁​ = m(x − x₁​)

`y - 3 = 1/3(x - 3)`

3(y − 3) = 1(x − 3)     ...[Multiplied by 3]

3y − 9 = x − 3

Let’s rearrange into the general form (Ax + By + C = 0):

x − 3y − 3 + 9 = 0

x − 3y + 6 = 0

Hence, the equation of the altitude through B is x − 3y + 6 = 0.