Question

A triangle has corners A, B, and C located at `(1 ,3 )`, `(9 ,5 )`, and `(6 ,2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

The endpoints are `(93/17, 70/17)`

length`=(9sqrt17)/17=2.18282" " "`units

Explanation:

There is a need to solve for the equation of line from A to B and the line from C perpendicular to line AB. When the two equations are obtained, solve simultaneously for their intersection point . This is the required point D.

Line from A(1, 3) to B(9, 5) using two-point form:

`(y-y_a)=((y_b-y_a)/(x_b-x_a))(x-x_a)`

`(y-3)=((5-3)/(9-1))(x-1)`

`y-3=1/4(x-1)`

`4y-12=x-1`
`x-4y=-11" " "`first equation

To solve for line from C perpendicular to line AB, use slope `=-4`
and point C(6, 2)

Using point-slope form:

`y-y_c=m(x-x_c)`

`y-2=-4(x-6)`

`y-2=-4x+24`

`4x+y=26" " "`second equation

Using first and second equation, solve simultaneously to find the unknow point D.

You should obtain `(x_d, y_d)=(93/17, 70/17)`

length`=sqrt((x_d-x_c)^2+(y_d-y_c)^2)`

length`=sqrt((93/17-6)^2+(70/17-2)^2)`

length`=sqrt((-9/17)^2+(36/17)^2)`

length`=sqrt((9^2/289+(9^2*4^2)/289)`

length`=9*sqrt(17/289)`

length`=9*sqrt(1/17)`

length`=(9sqrt17)/17=2.18282" " "`units

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