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ABC has vertices A(0, 6), B(4, 6), and C(1, 3). Sketch a graph of ABC and use it to find the orthocenter of ABC. Then list the steps you took to find the orthocenter, including any necessary points or slopes you had to derive.
To find the orthocenter of triangle ABC, we need to start by sketching the triangle.
First, plot the vertices A(0, 6), B(4, 6), and C(1, 3) on a coordinate plane.
Next, connect the three vertices to form triangle ABC.
Now we can find the slopes of the three sides of the triangle.
The slope of side AB can be found using the formula: m = (y2 - y1) / (x2 - x1)
For AB, (x1, y1) = (0, 6) and (x2, y2) = (4, 6)
mAB = (6 - 6) / (4 - 0) = 0 / 4 = 0
Similarly, we can find the slopes of sides BC and AC.
For BC, (x1, y1) = (4, 6) and (x2, y2) = (1, 3)
mBC = (3 - 6) / (1 - 4) = -3 / -3 = 1
For AC, (x1, y1) = (0, 6) and (x2, y2) = (1, 3)
mAC = (3 - 6) / (1 - 0) = -3 / 1 = -3
Next, we need to find the slopes of the altitudes of the triangle.
The slope of the altitude from A can be found by taking the negative reciprocal of the slope of BC.
mA_altitude = -1 / mBC = -1 / 1 = -1
The slope of the altitude from B can be found by taking the negative reciprocal of the slope of AC.
mB_altitude = -1 / mAC = -1 / -3 = 1/3
Finally, we need to find the intersection point of the altitudes, which is the orthocenter of the triangle.
To find the y-intercept of the altitude from A, we can use point-slope form y - y1 = m(x - x1) and plug in the point A(0, 6) and the slope mA_altitude.
y - 6 = -1(x - 0)
y = -x + 6
To find the y-intercept of the altitude from B, we can use point-slope form and plug in the point B(4, 6) and the slope mB_altitude.
y - 6 = 1/3(x - 4)
y = 1/3x + 6 - 4
y = 1/3x + 2
Now we have the equations of the two altitudes. To find the orthocenter, we need to find the intersection point of these two lines.
Setting the two equations equal to each other, we have:
-x + 6 = 1/3x + 2
Adding x to both sides and subtracting 2 from both sides:
2x = 4
Dividing both sides by 2:
x = 2
Plugging x = 2 into either equation, we can solve for the y-coordinate:
y = -x + 6
y = -2 + 6
y = 4
Therefore, the orthocenter of triangle ABC is located at (2, 4).
To summarize the steps:
1. Plot the vertices A(0, 6), B(4, 6), and C(1, 3) on a coordinate plane.
2. Connect the three vertices to form triangle ABC.
3. Find the slopes of sides AB, BC, and AC using the formula m = (y2 - y1) / (x2 - x1).
4. Find the slopes of the altitudes by taking the negative reciprocals of the slopes of the sides.
5. Use point-slope form to find the equations of the two altitudes.
6. Set the equations equal to each other and solve for x.
7. Plug x into either equation to find the y-coordinate of the orthocenter.
8. The orthocenter of triangle ABC is located at (2, 4).