Precalc If two points of a right triangle are given, how to find the third point using slopes.

Question

If two points of a right triangle are given, how to find the third point using slopes.

Answer 1

let (0,0) be one point and (4,0) be the other point, with (x,y) as the 3rd point of a triangle

slope from (0,0) to (x,y) is y/x

slope from (x,y) to (4,0) is y/(x-4)

if the first slope = 3 and 2nd slope = -1/3

then y/x= 3 and y/(x-4)=-1/3 are the two equations for the lines, 1st from (0,0) to (x,y) and 2nd from (x,y) to (0,4)

y=3x and y=-x/3+4/3

set them equal to find their intersection which is (x,y) the 3rd point, vertex, of the triangle.

3x =-x/3 +4/3

10x/3 = 4/3

x =2/5

y =3(2/5) =1.2

the 3rd vertex of the triangel is (1.2, 3.6)

whatever the initial 2 points, convert them so that they are on the x axis with one point at the origin

solve as above, then convert back from the converted solution back to the actual 3rd point

or if the slopes are negative inverses, as the following, then it's a also right triangle and

if the slopes are 1 and -1,

then the 3rd point (x,y) = (2,2), the midpont for the x coordinate, the average of the 2 other points (0+4)/2 = 2

y/x =1, y=x

y/(x-4) =-1, y=-x+4

x=-x+4

2x=4

x=4/2 =2

y=x =2

3rd point is (2,2)

but the right angle could place the 3rd point on the y axis, or directly vertically above the other point on the x axis.