what is the orthocenter of triangle with the vertices of X(-11,0),
Y(-11,-8), Z(-1,-4)?
Let Zz be the altitude from Z to XY. This is the line y = -4 since XY is vertical.
Let Xx be the altitude from X to YZ. This is a line through (-11,0) with slope -5/2.
(y-0)/(x+11) = -5/2
y = -5/2 x - 55/2
Where do the lines intersect?
-5/2 x - 55/2 = -4
-5x - 55 = -8
-5x = 47
x = -47/5
So, the orthocenter is at (-9.4,-4)
I'll let you verify by using the other altitude from Y to XZ.
Hey Steve,
Thanks....but how could it be -4?
never mind...i understand
To find the orthocenter of a triangle, we need to follow these steps:
Step 1: Find the slopes of the three sides of the triangle.
Step 2: Find the slopes of the altitudes (perpendiculars) to each side.
Step 3: Find the equations of the altitudes.
Step 4: Solve the system of equations formed by the three altitudes to find their intersection point, which is the orthocenter.
Let's go through each step:
Step 1: Find the slopes of the three sides of the triangle.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1).
Using the given vertices:
- The slope of line XY is (y2 - y1) / (x2 - x1) = (-8 - 0) / (-11 - (-11)) = -8 / 0 = undefined.
- The slope of line XZ is (y2 - y1) / (x2 - x1) = (-4 - 0) / (-1 - (-11)) = -4 / 10 = -2/5.
- The slope of line YZ is (y2 - y1) / (x2 - x1) = (-4 - (-8)) / (-1 - (-11)) = 4 / 10 = 2/5.
Step 2: Find the slopes of the altitudes (perpendiculars) to each side.
The slopes of perpendicular lines are negative reciprocals of each other. Therefore, the slopes of the altitudes will be reciprocals of the slopes of the corresponding sides.
The slope of the altitude from X to line YZ will be the negative reciprocal of the slope of line YZ, which is -5/2.
Similarly, the slope of the altitude from Y to line XZ will be the negative reciprocal of the slope of line XZ, which is 5/2.
The altitude from Z to line XY is already vertical, so its slope is undefined.
Step 3: Find the equations of the altitudes.
To find the equation of a line, we need its slope and a point it passes through.
Using the point-slope form of a line (y - y1) = m(x - x1):
The equation of the altitude from X to line YZ is: (y - 0) = (-5/2)(x - (-11)) => (y - 0) = (-5/2)(x + 11) => 2y = -5x - 55.
The equation of the altitude from Y to line XZ is: (y - (-8)) = (5/2)(x - (-11)) => (y + 8) = (5/2)(x + 11) => 2y = 5x + 27.
Step 4: Solve the system of equations formed by the three altitudes to find their intersection point, which is the orthocenter.
We have two equations: 2y = -5x - 55 and 2y = 5x + 27.
By setting them equal to each other, we can solve for the coordinates of the orthocenter.
-5x - 55 = 5x + 27
-10x = 82
x = -82 / 10
x = -41/5
Substituting this x-value back into either equation, we get:
2y = -5(-41/5) - 55
2y = 41 + 55
2y = 96
y = 96 / 2
y = 48
Therefore, the orthocenter of the triangle with vertices X(-11, 0), Y(-11, -8), and Z(-1, -4) is H(-41/5, 48).