Find the coordinates of the orthocenter of triangle ABC with vertices A(0,0), B(4,0), and C(4,2)
a. (4,0)
b. (2,1) ****
c. (4,2)
d. (2,0)
It’s A 4,0
Adjust the following procedure to fit your triangle
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To find the coordinates of the orthocenter of triangle ABC, we need to first find the equations of the altitudes of the triangle.
Step 1: Find the slope of line AB
The slope of line AB can be calculated using the formula: slope = (change in y) / (change in x).
The change in y is 0 since the y-coordinate of both A and B is 0.
The change in x is 4 - 0 = 4.
Therefore, the slope of line AB is 0/4 = 0.
Step 2: Find the equation of the altitude from vertex C
Since the slope of line AB is 0, the altitude from vertex C will be a vertical line passing through C(4,2).
The equation of a vertical line passing through the point (h, k) is x = h.
In this case, the equation of the altitude from vertex C is x = 4.
Step 3: Find the slope of line AC
The slope of line AC can be calculated using the formula: slope = (change in y) / (change in x).
The change in y is 2 - 0 = 2.
The change in x is 4 - 0 = 4.
Therefore, the slope of line AC is 2/4 = 1/2.
Step 4: Find the equation of the altitude from vertex B
The equation of the altitude from vertex B can be found using the formula: y - y1 = m(x - x1), where (x1, y1) is the coordinate of vertex B and m is the slope of line AC.
Using vertex B(4,0), the equation becomes: y - 0 = (1/2)(x - 4).
Simplifying, we get y = (1/2)x - 2.
Step 5: Solve for the intersection point of the altitudes
To find the coordinates of the orthocenter, we need to find the point where the equations of the altitudes intersect.
Solving the system of equations x = 4 and y = (1/2)x - 2 gives us x = 4 and y = (1/2)(4) - 2 = 2 - 2 = 0.
Therefore, the coordinates of the orthocenter of triangle ABC are (4, 0).
The correct answer is option a. (4,0).
To find the coordinates of the orthocenter of triangle ABC, you can follow the following steps:
Step 1: Determine the slopes of the lines containing the sides of the triangle.
First, find the slope of the line passing through points A(0,0) and B(4,0). The formula for finding the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1).
Slope AB = (0 - 0) / (4 - 0) = 0
Second, find the slope of the line passing through points A(0,0) and C(4,2).
Slope AC = (2 - 0) / (4 - 0) = 2/4 = 0.5
Step 2: Find the slopes of the perpendicular bisectors of the sides.
To find the slope of the perpendicular bisector of a line, take the negative reciprocal of the slope of the line. In this case, we need to find the slope of the perpendicular bisectors of AB and AC.
Slope of the perpendicular bisector of AB = -1 / 0 = Undefined
Slope of the perpendicular bisector of AC = -1 / 0.5 = -2
Step 3: Find the equations of the perpendicular bisectors.
To find the equation of a line given a slope (m) and a point (x1, y1), use the point-slope form: y - y1 = m(x - x1).
For AB: Since the slope is undefined, we know the perpendicular bisector is the vertical line passing through the midpoint of AB. The midpoint of AB can be found by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint AB = ((0 + 4) / 2 , (0 + 0) / 2) = (2,0)
The equation of the perpendicular bisector of AB is x = 2.
For AC: Using the slope (-2) and the point (2,1) (the midpoint of AC, found similarly by averaging the coordinates), we can find the equation of the perpendicular bisector.
Point-slope form: y - y1 = m(x - x1)
y - 1 = -2(x - 2)
y - 1 = -2x + 4
y = -2x + 5
Step 4: Find the intersection point of the perpendicular bisectors.
Solve the system of equations formed by the equations of the perpendicular bisectors to find the intersection point. In this case, solve the system: x = 2 and y = -2x + 5.
Substituting the value of x from the first equation into the second equation:
y = -2(2) + 5
y = -4 + 5
y = 1
Therefore, the intersection point of the perpendicular bisectors and the orthocenter of triangle ABC is (x, y) = (2, 1).
Based on the given options, the correct answer is b. (2,1).