In the triangle abc with vertices a 2 3

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Question 17 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.

Class X1 - Maths -Straight Lines Page 228

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Given, vertices A(2, 3) , B(4, -1) and C(1, 2)
Line AD ⊥ line BC
so, slope of AD × slope of BC = -1
Let slope of AD is m
slope of BC = (y2 - y1)/(x2 - x1)
= ( 2 +1)/(1 - 4) = 3/-3 = -1

m × (-1) = -1
m = 1

Hence, equation of AD by using
(y - y1) = m(x - x1)
here , (x1, y1) = (2, 3)
(y - 3) = m(x - 2)
(y - 3) = (x - 2)
y - x -1 = 0
x - y + 1 = 0

now, equation of BC by using
( y - y1) = (y2 - y1)/(x2 - x1) (x - x1)
here , (x1 , y1) = (4, -1) and (x2,y2) = (1,2)

(y + 1) ={(2 + 1)/(1 - 4)}(x - 4)
(y + 1) = (-1)(x - 4)
(y + 1) + (x - 4) = 0
x + y - 3 = 0

now, Length of AD = perpendicular distance from (2, 3) to the line BC
Length of AD = | 2 + 3 - 3 |/√(1²+1²)
= |2|/√2
= √2 unit .

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