Find the orthocenter of triangle with the following vertices (5,-2)(-1,2)(1,4)

Ortho center = (14/5, 1/5)

Step-by-step explanation:

Given, the vertices of the triangle,

A = (5, -2)

B = (-1, 2)

C = (1, 4)

Orthogonal center is the cross section of altitudes of the triangle.

Slope of AB

= y₂−y/x₂−x₁

= 2+2/-1-5

= -2/3

Altitude from C to AB is perpendicular to AB.

= Perpendicular slope of AB

= −1/Slope of AB

= 3/2

The equation of CF is given as, (F is the point on AB)

y – y₁ = m(x – x₁)

y - 4 = 3/2(x – 1)

2y – 8 = 3x - 3

3x - 2y = -5 ——— (1)

Slope of BC

= y₂–y/x₂–x₁

= 4–2/1+1

= 1

Slope of AD (AD is altitude)

Perpendicular slope of BC

= −1/Slope of BC

= −1

The equation of AD is given as,

y – y₁ = m(x – x₁)

y + 2 = -1(x – 5)

x + y = 3 ——– (2)

Subtracting equation (1) and 3 x (2),

(3x  -  2y  =  -5) - (3x  + 3y =  9)

= -5y = -14

y = 14/5

Substituting the value of y in equation (2),

X = 3 – 14/5 = 1/5

Ortho center = (14/5, 1/5).