# Thread: problem dealing with P.O.I. and equation of a line in 2d(impossible!!!)

1. ## problem dealing with P.O.I. and equation of a line in 2d(impossible!!!)

hello, i recently had some homework to do for school and i came across a problem in the text that I couldn't solve so here is the problem and i will really appreciate it if somebody could solve it:

the line segment joining A(2,3) to B(9,2) is the hyptenuse of a right triangle. The third vertex, C, lies on the line with these parametric equations:

x = 2 + 2t
y = 8 -t
determine the coordinates of C.

thanks a lot for any help!

2. Determine the line where C is by eliminating the parameter t ....so that you will be dealing with x and y only, as the coordinates are in x and y only.

y = 8 -t
So, t = 8 -y
Then,
x = 2 +2t
x = 2 +2(8 -y)
x = 2 +16 -2y
x = 18 -2y
2y = -x +18
y = (-1/2)x +9 ----------the line where C is.

If AB is the hypotenuse of the right triangle with C as the right angle, then, by Pythagorean theorem,
(AB)^2 = (AC)^2 +(BC)^2 ----------(i)

Let C be point C(x,y).
Umm, so it is simpler to express the x in terms of y.
Since x = 18 -2y, so,
C = C(18-2y,y)

Then in (i),
(9-2)^2 +(2-3)^2 = [((18-2y)-2)^2 +(y-3)^2] +[(18-2y)-9)^2 +(y-2)^2]
Expand, simplify and solve for y.

Then, x = 18 -2y.

You should get two possible C's......(6,6) and (8,5).