Q.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 $A$.

Concept: Coordinate Geometry — altitude from a vertex is perpendicular to the opposite side.

Step 1 — Slope of BC

$B(4, -1)$, $C(1, 2)$

Slope $m_{BC} = \frac{2 - (-1)}{1 - 4} = \frac{3}{-3} = -1$

Step 2 — Slope of altitude from A

Perpendicular slope $m = 1$ (negative reciprocal of $-1$).

Equation through $A(2, 3)$:

$y - 3 = 1(x - 2) \implies y = x + 1$

Step 3 — Foot of altitude (intersection with BC)

Line BC: slope $-1$ through $B(4, -1)$: $y + 1 = -1(x - 4) \implies y = -x + 3$

Solve $y = x + 1$ and $y = -x + 3$:

$x + 1 = -x + 3 \implies 2x = 2 \implies x = 1$, then $y = 2$ …

Why this works — the idea

An altitude in a triangle is a line from a vertex perpendicular to the opposite side. So the altitude from A is the line through A that is perpendicular to BC. Its length is simply the perpendicular distance from A to line BC — that's the shortest distance from A to side BC.

We'll do this in two clean parts: first find the equation of the altitude (the line), then find its length.

Step-by-step solution

1. Find the slope of BC

Points: $B(4, -1)$ and $C(1, 2)$.

Slope of BC:

2. Slope of the altitude from A

Since the altitude is perpendicular to BC, its slope $m$ satisfies:

So:

3. Equation of the altitude through A(2, 3)

Using point-slope form:

Simplify:

So the altitude from A is the line $x - y + 1 = 0$.

4. Length of the altitude

The length of the altitude is the perpendicular distance from A to line BC. First, find the equation of BC.

Using points B(4, -1) and C(1, 2):

So: …