Question

A(0,0) B(0,-2) C(-3,0) Find the orthocenter?

Answer 1

Ortho center:  Ortho center of a triangle is the point of intersection of all the attitudes of the triangle.

The vertices of triangle are A (0, 0) B(0, - 2) and C(- 3, 0)

Step 1: find the slope of  sides of triangle.

AB , CF are perpendicular.

We know that perpendicular lines slopes are reciprocal to each other.

(x₁, y₁) = A(0, 0) and (x₂, y₂) = B(0, - 2).

Slope of AB (m) = [(y₂ - y₁)/(x₂ -x₁)]

m = (- 2 - 0)/(0 - 0)

m  = undefined.

Vertical line slope is undefined.

Slope of horizontal line = 0.

Slope of CF = 0.

Substitute m = 0, C (- 3, 0) = (x₁, y₁)  in point - slope form y - y₁ = m(x - x₁)

y - 0 = 0[x - (- 3)]

y = 0 ---> (1)

BC, AD are perpendicular.

B(0, - 2) = (x₁, y₁)  and (x₂, y₂) = C(- 3, 0)

Slope of BC = (0 + 2)/(- 3 - 0)

m = -2/3

Substitute m = - 2/3, A(0, 0) = (x₁, y₁)  in point - slope form y - y₁ = m(x - x₁)

y - 0 = (-2/3)(x - 0)

y = -2x/3 ---> (2)

 

Step 2: Solving two altitude CF, AD equations.

Equate the y forms.

-2x/3 = 0

- 2x = 0

x = 0

Substitute in x = 0 in y = -2x/3

y = 0

Orthocenter (x,y) = (0, 0).

We can determine orthocenter in additional method.

Plot the points and connect the sides of traingle in a coordinate plane.

ABC is a right angle triangle, the orthocenter is the is the vertex which is the right angle.

Orthocenter is (0, 0).