Question

A triangle has corners A, B, and C located at `(7 ,6 )`, `(4 ,3 )`, and `(5 ,8 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

`6,7),(5,8),2sqrt2`

Explanation:

`"the altitude from C meets AB the side opposite at right"`
`"angles"`

`"let D be the point on AB where the altitude intersects AB"`

`m_(AB)=(6-3)/(7-4)=3/3=1`

`"hence "m_(CD)=-1/m_(AB)=-1`

`color(blue)"equation of AB"`

`"using "m=1" and "(a,b)=(4,3)" then"`

`y-3=x-4rArry=x-1to(1)`

`color(blue)"equation of CD"`

`"using "m=-1" and "(a,b)=(5,8)" then"`

`y-8=-(x-5)`

`y=-x+13to(2)`

`"solving "(1)" and "(2)`

`x-1=-x+13rArrx=7`

`y=7-1=6`

`"coordinates of D "=(7,6)`

`"thus the endpoints of the altitude are "(7,6)" and "(5,8)`

`d=sqrt((5-7)^2+(8-6)^2)=sqrt(4+4)=sqrt8=2sqrt2`

`"length of altitude "=2sqrt2`