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

The vertices of the triangle are A ( 2,3 ) , B ( 4,−1 ) , C ( 1,2 ) respectively.

Let the perpendicular intersect at point D on side BC .

The formula for the slope of a line passes through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let m 1 be the slope of the line segment which passes through the points B ( 4,−1 ) , C ( 1,2 ) .

Substitute the value of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 4,−1 ) and ( 1,2 ) respectively in equation (1).

m 1 = 2+1 1−4 = 3 −3 =−1 (2)

Let m 2 be the slope of the perpendicular to the line.

The product of the slope of two lines perpendicular to each other is −1 .

m 1 ⋅ m 2 =−1 (3)

Substitute the value of m 1 from equation (2) to equation (3) respectively.

−1⋅ m 2 =−1 m 2 = −1 −1 =1

The formula for the equation of the line having slope m passes through the point ( x 1 , y 1 ) is given by,

( y− y 1 )=m( x− x 1 ) (4)

Substitute the values of m as 1 and ( x 1 , y 1 ) as ( 2,3 ) in equation (4).

( y−3 )=1( x−2 ) y−3=x−2 x−y−2+3=0 x−y+1=0

The formula for the equation of line passing through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

( y− y 1 )= y 2 − y 1 x 2 − x 1 ( x− x 1 ) (5)

The equation of line passing through points ( 4,−1 ) and ( 1,2 ) is obtained by substituting the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( 4,−1 ) and ( 1,2 ) respectively.

( y+1 )= 2+1 1−4 ( x−4 ) ( y+1 )= −3 3 ( x−4 ) ( y+1 )=−1( x−4 ) y+1=−x+4

Further simplifying the above expression.

x+y+1−4=0 x+y−3=0

The general form of the equation of line is given by,

Ax+By+C=0 (6)

Compare the above expression with the general form of equation of line from equation (6).

A=1, B=1, C=−3 (7)

The formula for the perpendicular distance d of a line Ax+By+C=0 from a point ( x 1 , y 1 ) is given by,

d= | A x 1 +B y 1 +C | A 2 +B 2 (8)

Substitute the value of ( x 1 , y 1 ) as ( 2,3 ) and the values of A , B ,and C from equation (7) to equation (8).

d= | 1×2+1×3−3 | 1 2 + 1 2 = | 2 | 2 = 2 2 = 2  units

Thu, the equation of the altitude is x−y+1=0 and length of altitude is 2  units .