how do you find the orthocenter of a triangle with the vertices A(4,0) B(-2,4)
C(0,6)?
Please explain to me how you get x after doing point slope form for the two slopes.
the orthocentre is the intersection of the altitudes.
slope AB = (4-0)/(-2-4) = -2/3
so slope of altutude from C is 3/2 , and y-intercept is 6
equation of altitude from C is y = (3/2)x + 6
slope AC = 6/-4 = -3/2
slope of altitude from B is 2/3
equation: y = (2/3)x + b
but B(-2,4) lies on it, so
4 = (2/3)(-2) + b
12 = -4 + 3b
b = 16/3 , so altitude from B to AC is y = (2/3)x + 16/3
the orthocentre is the intersection of these two lines
(2/3)x + 16/3 = (3/2)x + 6
multiply by 6
4x + 32 = 9x + 36
-4 = 5x
x = -4/5
y = (3/2)(-4/5) + 6 = 24/5
The orthocentre is (-4/5, 24/5)
To find the orthocenter of a triangle with given vertices, you need to determine the point where the altitudes of the triangle intersect.
To find the orthocenter, you will first need to determine the equations of the altitudes. An altitude of a triangle is perpendicular to the opposite side and passes through the opposite vertex.
First, let's find the slopes of the lines passing through each side of the triangle. To find the slope of a line, you use the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
For side AB:
slope_AB = (y₂ - y₁) / (x₂ - x₁)
= (4 - 0) / (-2 - 4)
= 4 / -6
= -2/3
For side BC:
slope_BC = (y₂ - y₁) / (x₂ - x₁)
= (6 - 4) / (0 - (-2))
= 2 / 2
= 1
Now, let's find the equations of the lines passing through each side. To find the equation of a line using point-slope form, you use the formula:
y - y₁ = m(x - x₁)
For side AB, using point A (4, 0):
y - 0 = (-2/3)(x - 4)
y = (-2/3)x + (8/3)
For side BC, using point B (-2, 4):
y - 4 = 1(x + 2)
y = x + 6
Now, to find the intersection point of the two altitudes (the orthocenter), you need to solve the system of equations formed by the two altitude lines:
(-2/3)x + (8/3) = x + 6
To solve for x, you can start by isolating x on one side of the equation:
(-2/3)x - x = 6 - (8/3)
(-8/3)x = 6 - (8/3)
Now, you can simplify the equation:
(-8/3)x = (18/3) - (8/3)
(-8/3)x = 10/3
To remove the negative sign and simplify the equation further, multiply both sides of the equation by -3/8:
x = (10/3) * (-3/8)
x = -10/8
x = -5/4
So, the x-coordinate of the orthocenter is -5/4.
I hope this explanation helps you understand the process of finding the x-coordinate after using point-slope form for the slopes of the triangle's sides.