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 AD be the altitude of triangle ABC from vertex A. Accordingly, AD ⊥ BC

The slope of BC = (2 + 1) / (1 - 4) = 3/(-3) = -1.

Since AD is perpendicular to BC, the slope of AD = -1 / (slope of BC) = -1/(-1) = 1.

The equation of the line passing through point (2, 3) and having a slope of 1 (equation of AD) is

⇒ ( y - 3) = 1(x - 2)

⇒ y - 3 = x - 2

⇒ y - x = 1

Therefore, equation of the altitude from vertex A = y - x = 1

Length of AD = Length of the perpendicular from A(2, 3) to BC

The equation of BC is

⇒ (y + 1) = (-1) (x - 4)

⇒ y + 1 = - x + 4

⇒ x + y - 3 = 0 ....(1)

The perpendicular distance (d) of a line Ax + By + C = 0 from a point (x\(_1\), y\(_1\)) is given by

d = |Ax\(_1\) + By\(_1\) + C|/√A² + B²

On comparing equation (1) to the general equation of line Ax + By + C = 0 , we obtain A = 1, B = 1 and C = -3.

Length of AD = |1 × 2 + 1 × 3 - 3|/√1² + 1² = 2/√2 = √2 units

Thus, the equation and length of the altitude from vertex A are y - x = 1 and √2 units respectively

NCERT Solutions Class 11 Maths Chapter 10 Exercise 10.3 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.

Summary:

The equation and length of the altitude from vertex A are y - x = 1 and √2 units respectively