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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.
Solution
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`
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