Find the orthocenter of triangle ABC.
A (3,5) B (9,5) C (5,3)
I think it is (9, 5).
The correct answer would be (5,1)
To find the orthocenter of a triangle, you need to follow these steps:
Step 1: Find the slopes of the lines passing through each pair of sides of the triangle.
Step 2: Perpendicular lines have slopes that are negative reciprocals of each other. Find the slopes of the perpendicular lines passing through each side of the triangle.
Step 3: Use the slopes and the coordinates of the vertices of triangle ABC to find the equations of the perpendicular lines.
Step 4: Solve the system of equations formed by the equations of the perpendicular lines to find the intersection point, which is the orthocenter.
Let's go through the process step by step.
Step 1:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
For side AB:
slope_AB = (5 - 5) / (9 - 3) = 0 / 6 = 0
For side BC:
slope_BC = (3 - 5) / (5 - 9) = -2 / -4 = 1/2
For side CA:
slope_CA = (3 - 5) / (5 - 3) = -2 / 2 = -1
Step 2:
The slopes of perpendicular lines to lines with slopes m and n are negative reciprocals of each other. So, the slopes of the perpendicular lines passing through the sides of the triangle are:
For side AB: -1/0 (undefined slope since the original slope is 0)
For side BC: -2
For side CA: -1/2
Step 3:
Now we have the slopes of the perpendicular lines and the coordinates of the vertices. We can use the point-slope formula to write the equations of the perpendicular lines.
For side AB, we have an undefined slope (perpendicular to a horizontal line). So, the equation of the perpendicular line passing through A is simply x = 3.
For side BC, we have a slope of -2. Using the point-slope formula with the point (9, 5), we can write the equation of the line passing through B as:
y - 5 = -2(x - 9)
Simplifying the equation:
y - 5 = -2x + 18
y = -2x + 23
For side CA, we have a slope of -1/2. Using the point-slope formula with the point (5, 3), we can write the equation of the line passing through C as:
y - 3 = (-1/2)(x - 5)
Simplifying the equation:
y - 3 = (-1/2)x + 5/2
2y - 6 = -x + 5
2y = -x + 11
y = (-1/2)x + 11/2
Step 4:
We have three equations for the perpendicular lines:
x = 3
y = -2x + 23
y = (-1/2)x + 11/2
We can solve this system to find the intersection point, which is the orthocenter.
By substituting x = 3 in the other two equations:
For y = -2x + 23:
y = -2(3) + 23
y = 23 - 6
y = 17
For y = (-1/2)x + 11/2:
y = (-1/2)(3) + 11/2
y = -3/2 + 11/2
y = 8/2
y = 4
Therefore, the coordinates of the orthocenter of triangle ABC are (3, 17).
So, the orthocenter is not (9, 5) as you suggested.