Question

Find the coordinates of the orthocenter of △ A B C with vertices A(-3,3), B(-1,7), and C(3,3). You must show all of your steps.

Answer 1

( -1,-5 )

Explanation:

We have the co-ordinates A( -3,3 ), B( -1,7 ) and C( 3,3 ).

We will find the orthocenter using below steps:

1. First, we find the equations of AB and BC.

The general form of a line is y=mx+b where m is the slope and b is the y-intercept.

Using the formula of slope given by
`m=(y_(2)-y_(1))/(x_(2)-x_(1) )`, we will find the slope of AB and BC.

Now, slope of AB is
`m=(7-3)/(-1+3)` i.e.
`m=(4)/(2)` i.e.
`m=2`.

Putting this 'm' in the general form and using the point B( -1,1 ), we get the y-intercept as,

y = mx + b i.e. 1 = 2 × (-1) + b i.e. b = 3.

So, the equation of AB is y = 2x + 3.

Also, slope of BC is
`m=(3-7)/(3+1)` i.e.
`m=(-4)/(4)` i.e.
`m=-1`.

Putting this 'm' in the general form and using the point B( -1,1 ), we get the y-intercept as,

y = mx + b i.e. 1 = (-1) × (-1) + b i.e. b = 0.

So, the equation of BC is y = -x.

2. We will find the slope of line perpendicular to AB and BC.

When two lines are perpendicular, then the product of their slopes is -1.

So, slope of line perpendicular to AB is
`m * 2 = -1` i.e.
`m=(-1)/(2)`

So, slope of line perpendicular to BC is
`m * (-1) = -1` i.e. m = 1.

3. We will now find the equations of line perpendicular to AB and BC.

Using the slope of line perpendicular to AB i.e.
`m=(-1)/(2)` and the point opposite to AB i.e. C( 3,3 ), we get,

y = mx+b i.e.
`3=(-1)/(2) * 3 + b` i.e.
`b=(9)/(2)`

So, the equation of line perpendicular to AB is
`y=(-x)/(2) +(9)/(2)`

Again, using the slope of line perpendicular to BC i.e. m = 1 and the point opposite to BC i.e. A( -3,3 ), we get,

y = mx + b i.e. 3 = 1 × -3 + b i.e. b = 6.

So, the equation of line perpendicular to BC is y = x+6

4. Finally, we will solve the obtained equations to find the value of ( x,y ).

As, we have y = x+6 and
`y=(-x)/(2)+(9)/(2)`

This gives,
`y=(-x)/(2)+(9)/(2)` →
`x+6=(-x)/(2) +(9)/(2)` → 2x+12 = -x+9 → 3x = -3 → x = -1.

So, y = x+6 → y = -1+6 → y=5.

Hence, the orthocenter of the ΔABC is ( -1,5 ).