Question

i need help on finding the orthocenter of a triangle, i got the x right, x=6 but I got the Y value wrong, it's supposed to be y=4 but i got y=8 as my answer. the question is find the coordinates of the orthocenter of Triangle ABC with vertices A(2,6), B(8,6), and C(6,2). what i did to find the Y value was find the slope of BC then use that value and the coordinates of A and substituted them into an equation y-6=1/2(x-2), i solved it and got y = 1/2x+5. i then substituted my x value that i previously found into it to find Y , y=1/2(6)+5 . i got 8 as the Y value but it's wrong, can someone tell me what i missed?

Answer 1

See the attached

Explanation:

When in doubt, draw a diagram.

The orthocenter of this acute triangle will be within its bounds. That should tell you right away that the y-coordinate of it will not be 8, but must be between 2 and 6.

The line perpendicular to BC through A must have a y-intercept greater than the y-coordinate of A, so cannot be 5. Whatever it is, the y-coordinate of the orthocenter will be less, so again, your answer fails the reasonableness test.

The perpendicular line to BC through A is ...

... y = (-1/2)(x -2) +6 = -x/2 +7 . . . . . . looks like you had a sign error in (-1/2)(-2)

The intersection of that line and x=6 is ...

... y = -6/2 +7 = 4