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.
9 years and nobody knows
whats the answer? :( its important that i do well on this test i just need to know the method....
To sketch the graph of ABC, let's plot the given points A(0, 6), B(4, 6), and C(1, 3) on a coordinate plane.
First, plot point A(0, 6) at the origin of the coordinate plane.
Next, plot point B(4, 6) on the x-axis, 4 units to the right of A.
Finally, plot point C(1, 3) somewhere below and to the right of A and B.
Now, join points A, B, and C to form triangle ABC.
To find the orthocenter of ABC, you need to follow these steps:
1. Find the slopes of the sides of triangle ABC.
- Slope of AB = (6 - 6) / (4 - 0) = 0
- Slope of BC = (6 - 3) / (4 - 1) = 1
- Slope of CA = (3 - 6) / (1 - 0) = -3
2. Find the slopes of the altitudes, which are perpendicular to the corresponding side of the triangle.
- Slope of the altitude from A is the negative reciprocal of the slope of BC, which is -1.
- Slope of the altitude from B is the negative reciprocal of the slope of CA, which is 1/3.
- Slope of the altitude from C is the negative reciprocal of the slope of AB, which is undefined (vertical line).
3. Find the equations of the altitudes using the point-slope form.
- Equation of the altitude from A: y - 6 = -1(x - 0) --> y = -x + 6
- Equation of the altitude from B: y - 6 = (1/3)(x - 4) --> y = (1/3)x + (14/3)
- Equation of the altitude from C: x = 1 (vertical line passing through C)
4. Solve the system of equations to find the point of intersection.
- By solving, we get the point of intersection as (-2, 8).
Therefore, the orthocenter of triangle ABC is at the point (-2, 8).
pick a vertex.
find the slope of the perpendicular to the opposite side.
find the equation of the line with that slope, going through the vertex.
pick another vertex and repeat
find the intersection of the two lines.
That's the orthocenter.
you copy that from a different person
It’s 21
Method:
Find the equation of two of those altitudes.
Solve the two equations to find their intersection point.
How?
Make a rough sketch
Pick any point and find the slope of the opposite side.
The slope of the altitude to that side is the negative reciprocal of the slope of that side.
Now you have the slope and a point on that line, find the equation for the line.
Repeat the above for a second altitude, solve the two equations.
PS, just noticed how nice your points are. One of the lines is a horizontal line, so the altitude from (1,3) to that line is x = 1