Find the equation of the altitude of the triangle

Question

A(1,-5),B(2,2) and C(-2,4) are the vertices of triangle ABC . Find the equation of the altitude of the triangle through B.

The answer given at the back of my book is 3y=x+4. But how is it coming.

Answer 1

A(1,-5),B(2,2) and C(-2,4) are the vertices of triangle ABC . Find the equation of the altitude of the triangle through B.

The answer given at the back of my book is 3y=x+4. But how is it coming.

The altitude of the point B is a line perpinficular to the base line AC, and passing through the point B(2,2).

Find the slope of the base line AC

Hence find the slope of any line perpindicular to it.

This line will now pass through the point B(2,2)

You have a line with a slope that passes through a fixed point.

Hence you can find the equation of this line.

Answer 2

Assuming that the longest side is the base then AC is the longest side. The slope of this side is the difference of the y values divided by the difference of the x values: (4-(-5))/(-2-1)=-9/3=-3. The altitude has a slope which is the negative inverse of this, that is, 1/3. So we know the altitude has an equation of the form y=x/3+b where b is to be found. Since AC is the base, the vertex B at (2,2) lies on the altitude line, so we plug its coordinates in:

2=2/3+b, and b=2-2/3=4/3. The equation of the altitude is y=x/3+4/3 which is better written 3y=x+4.