find the orthocenter of the triangle whose vertices are (5,-2),(-1,2) and (1,4)
Answers
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*(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)
Answer:
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