Given tri ABC where A(4,-6) B, (-8,2)
C (-4,8)
A. Write the equation of the altitudes of triangle AC
b. Determine the point of concurrency of the altitudes
c. What is this point of concurrency called?
a. To find the equation of the altitude of triangle AC, we first need to determine the slope of line BC, which is perpendicular to the altitude. The slope of BC can be found using the formula:
slope_BC = (y2 - y1) / (x2 - x1)
Given points B(-8, 2) and C(-4, 8), the slope of BC is:
slope_BC = (8 - 2) / (-4 - (-8)) = 6 / 4 = 3/2
Since the altitude is perpendicular to BC, the slope of the altitude is the negative reciprocal of the slope of BC. Therefore, the slope of the altitude is:
slope_altitude = -2/3
Now, we can use the point-slope form of the equation of a line to write the equation of the altitude.
Using point A(4, -6), the equation of the altitude is:
y - y1 = m(x - x1)
y - (-6) = (-2/3)(x - 4)
Simplifying the equation, we have:
y + 6 = (-2/3)(x - 4)
b. To determine the point of concurrency of the altitudes, we need to find the intersection point of all three altitudes. So, we should find the equations of the other two altitudes first and then solve the system of equations to find the point of concurrency.
We already found the equation for altitude AC above:
y + 6 = (-2/3)(x - 4)
Next, we need to find the equations for the other two altitudes. Let's find the equation for altitude AB:
To find the slope of line AC, we use the formula:
slope_AC = (y2 - y1) / (x2 - x1)
Given points A(4, -6) and C(-4, 8), the slope of AC is:
slope_AC = (8 - (-6)) / (-4 - 4) = 14 / (-8) = -7/4
Since the altitude is perpendicular to AC, the slope of the altitude is the negative reciprocal of the slope of AC. Therefore, the slope of the altitude is:
slope_altitude_AB = 4/7
Using point B(-8, 2), the equation of altitude AB is:
y - y1 = m(x - x1)
y - 2 = (4/7)(x - (-8))
Simplifying the equation, we have:
y - 2 = (4/7)(x + 8)
Now, let's find the equation for altitude BC:
To find the slope of line AB, we use the formula:
slope_AB = (y2 - y1) / (x2 - x1)
Given points A(4, -6) and B(-8, 2), the slope of AB is:
slope_AB = (2 - (-6)) / (-8 - 4) = 8 / (-12) = -2/3
Since the altitude is perpendicular to AB, the slope of the altitude is the negative reciprocal of the slope of AB. Therefore, the slope of the altitude is:
slope_altitude_BC = 3/2
Using point C(-4, 8), the equation of altitude BC is:
y - y1 = m(x - x1)
y - 8 = (3/2)(x - (-4))
Simplifying the equation, we have:
y - 8 = (3/2)(x + 4)
c. The point of concurrency of the altitudes is called the orthocenter of the triangle. It is denoted by the letter H. To find the coordinates of the orthocenter, we need to solve the system of equations formed by the three altitude lines.
By solving the three equations obtained above, we can find the coordinates of the point of concurrency, which is the orthocenter of triangle ABC.