Question

In the triangle ABC with vertices A (2, 3), B (4, –1) and C (1, 2), find the equation and length of altitude from the vertex A.

Answer 1

Let AM be perpendicular to line BC.

(i) Slope of line BC

= `("y"_2 - "y"_1)/("x"_2 - "x"_1)`

= `(2 + 1)/(1 - 4)`

= `3/ (-3)`

= −1

AM ⊥ BC,

∴ Slope of perpendicular AM = `(-1)/"m"`

= `(-1)/(-1)`

= 1

Line AM passes through point A and slope = 1.

∴ equation of AM

y – y₁ = m(x – x₁)

y – 3 = 1(x – 2)

or x – y + 1 = 0

(ii) Equation of line BC passing through points B(4, –1) and C(1, 2)

`"y"- "y"_1 = ("y"_2 - "y"_1)/("x"_2 - "x"_1)("x" - "x"_1)`

y + 1 = `(2 + 1)/(1 - 4) ("x" - 4)`

= `3/(-3) ("x" - 4)`

= −x + 4

x + y − 3 = 0

∴ Length of perpendicular AM from point A to BC

= `(2 + 3 -3)/sqrt(1^2 + 1^2)` ..........`[∵ "d" = ("ax"_1 + "by"_1 + "c")/sqrt("a"^2 + "b"^2)]`

= `2/sqrt2`

= `sqrt2`