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

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

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

Answer»

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).