Question

A triangle has corners A, B, and C located at `(2 ,9 )`, `(1 ,4 )`, and `(6 , 5 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

The endpoints are `(18/13,77/13)` and `(6,5)` and length of the altitude going through corner C is `4.707`

Explanation:

The slope of line `AB` is `(4-9)/(1-2)=(-5)/(-1)=5` and equation of `AB` is `(y-4)=5(x-1)`

i.e. `5x-y=1` ................................(1)

Hence if `CD_|_AB`, with `D` on `AB`, slope of `CD` will be `(-1)/5=-1/5`

and equation of `CD` is `(y-5)=-1/5(x-6)`

or `5y-25=-x+6` i.e. `x+5y=31` ................................(2)

Solving (1) and (2) gives us the coordinates of `D`. For tis putting `y=5x-1` from (1) in (2) gives

`x+25x-5=31` or `26x=36` i.e. `x=36/26=18/13`

and hence `y=5xx18/13-1=77/13`

and coordinated of `D` are `(18/13,77/13)`

and `CD=sqrt((6-18/13)^2+(5-77/13)^2)`

= `sqrt((60/13)^2+(-12/13)^2)`

= `1/13sqrt(3600+144)=sqrt3744/13=61.188/13=4.707`