Question

A triangle has corners A, B, and C located at `(3 ,1 )`, `(6 ,4 )`, and `(9 ,8 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

1. First find the line `AB`
2. Then find the line `l` perpendicular to `AB` (the line of altitude) such that it intersects with point `C`
3. Then find the point `D` at which `l` intersects with `AB`
4. Then use the distance formula with the points `C` and `D` to get the distance of the altitude.

See if you can do it step by step. If not, keep on reading.

Step 1

Find the equation for `AB`

`y-y_1=m(x-x_1)` (point slope formula)

`m = (4-1)/(6-3) = 1` (finding the slope using the points `A` and `B`)

`y-1=x-3` (plugging `m` and A back into the equation, you could choose A or B and it would be the same)

Segment `AB` lies on the line `y=x-2`

Step 2

Find line `l` that is perpendicular to `AB` and intersects with `C`.

`y = mx + b` (start with an empty line)

`y = -1x + b` (negative reciprocal slope of `AB` for perpendicular line)

`8 = -9 + b` (substitute in point `C` to find a line that intersects with `C`)

`b = 17` (solve)

Now that we have `b` we can list the equation.

`l`, our altitude, has the equation `y = -x+17`

Step 3

Find point `D` where our altitude `l` and base `AB` intersect

`y = -x+17` (`l`)
`y=x-2` (`AB`)

So `-x+17=x-2`
`19=2x`
`x = 9.5`
`y = 7.5` (by plugging `x` back into one of the equations)

So #D = (9.5,7.5)

Step 4

Find the distance of `CD`

`sqrt((9-9.5)^2+(8-7.5)^2)`

`= sqrt(2)/2`

Hm.. the answer doesn't look quite right. Whoops.