In the triangle ABC with vertices A(2, 3), B(4, -1) and C(1, 2) find the equation and the length of the altitude from the vertex A.
Given:
A(2, 3), B(4, -1) and C(1, 2).
To find:
The equation and the length of the altitude from the vertex A.
Concept Used:
Distance of a point from a line.
Explanation:
Equation of side BC:
⇒ x + y – 3 = 0
The equation of the altitude that is perpendicular to x + y – 3 = 0 is x – y + λ = 0.
Line x – y + λ = 0 passes through (2, 3).
∴ 2 – 3 + λ = 0
⇒ λ = 1
Thus, the equation of the altitude from the vertex A (2, 3) is x – y + 1 = 0.
Let d be the length of the altitude from A (2, 3).
d
⇒ d
Hence, the required distance is.