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...

Answer 1

To find the orthocenter of triangle ABC, we need to find the point where the altitudes of the triangle intersect.

Step 1: Sketch the triangle ABC
Using the given coordinates, we can plot the points on a graph:
A(0, 6)
B(4, 6)
C(1, 3)

Step 2: Draw the altitudes
The altitude of a triangle is a perpendicular line from one vertex to the opposite side. We need to find the equations of the three altitudes.

To find the equation of the altitude from A to BC:
The slope of line BC can be found using the coordinates of points B and C. The slope formula is given by:
slope of BC (mBC) = (y2 - y1) / (x2 - x1)
mBC = (6 - 3) / (4 - 1)
mBC = 3/3
mBC = 1

Since the altitude from A is perpendicular to BC, the slope of this altitude will be the negative reciprocal of mBC.
Slope of altitude from A to BC = -1/mBC = -1/1 = -1

To find the equation of the altitude from A(0, 6) with slope -1:
Using the point-slope form of a linear equation:
y - y1 = m(x - x1)
y - 6 = -1(x - 0)
y - 6 = -x
y = -x + 6 (equation of altitude from A to BC)

Similarly, we can find the equations of the altitudes from B and C:
Slope of altitude from B to AC = -1,
Equation: y = 6 (horizontal line passing through B)

Slope of altitude from C to AB = -1/2,
Equation: y = -x + 9 (using point-slope form with point C(1, 3) and slope -1/2)

Step 3: Find the intersection point of the altitudes
To find the orthocenter, we need to find the point where all three altitudes intersect. We can do this mathematically by solving the system of equations formed by the three altitudes.
Solving y = -x + 6 and y = 6, we get:
-x + 6 = 6
-x = 0
x = 0

Substituting x=0 back into any of the equations, we get y = 6.

Therefore, the orthocenter of triangle ABC is H(0, 6).

Step 4: Plot the orthocenter on the graph
Plot the point H(0, 6) on the graph.

Step 5: Connect the orthocenter to the vertices
Draw the lines HA, HB, and HC to complete the triangle HBC.

The final graph should show triangle ABC with its altitudes and orthocenter H(0, 6).