Question

Triangle ABC has vertices A (0,10) B (4,10) and C (-2,4) find the orthocenter of triangle ABC

Answer 1

(-2, 12) is the orthocenter of triangle ABC

Answer 2

Hence the orthocenter is (-2,12)

Explanation:

We need to find the Orthocenter of ΔABC with vertices A(0,10) , B(4,10) and C(-2,4).

" Orthocenter of a triangle is a point of intersection, where three altitudes of a triangle connect ".

Step 1 : Find the perpendicular slopes of any two sides of the triangle.

Step 2 : Then by using point slope form, calculate the equation for those two altitudes with their respective coordinates.

Step 1 : Given coordinates are: A(0,10) , B(4,10) and C(-2,4)

Slope of BC =
`(y_(2)-y_(1))/(x_(2)-x_(1))= (4-10)/(-2-4)=(-6)/(-6)=1`

Perpendicular Slope of BC = -1

( since for two perpendicular lines the slope is given as:
`m_(1)* m_(2)=-1`

where
`m_(1),m_(2)` are the slope of the two lines. )

Slope of AC =
`(y_2-y_1)/(x_2-x_1)=(4-10)/(-2-0)=(-6)/(-2)=3`

Perpendicular Slope of AC=
`(-1)/(3)`

Step 2 : Equation of AD, slope(m) = -1 and point A = (0,10)

`y - y_(1) = m* (x-x_(1))`

`y - 10 = -1(x - 0)\\y - 10 = -x \\x + y = 10---------------(1)`

Equation of BE, slope(m) =\dfrac{-1}{3} and point B = (4,10)

`y - y_1 = m(x - x_1)`

`y - 10= (-1)/(3) * (x - 4)`

`3y-30=-x+4`

`x+3y =34`----------(2)

Solving equations (1) and (2), we get

(x, y) = (-2,12)

Hence, the orthocenter is (-2,12).