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.
I don't get it at all
To sketch the graph of triangle ABC, we plot the given points A(0, 6), B(4, 6), and C(1, 3) on a coordinate plane.
Now let's find the orthocenter of triangle ABC. The orthocenter is the point of intersection of the altitudes of a triangle, where an altitude is a perpendicular line segment from a vertex to the line containing the opposite side.
To find the orthocenter, we first need to find the equations of the three altitudes of triangle ABC.
Step 1: Determine the slopes of the sides of the triangle.
- The slope of line AB can be found using the formula: slope = (y2 - y1) / (x2 - x1).
slope_AB = (6 - 6) / (4 - 0) = 0 / 4 = 0.
- The slope of line BC:
slope_BC = (3 - 6) / (1 - 4) = -3 / -3 = 1.
- The slope of line AC:
slope_AC = (3 - 6) / (1 - 0) = -3 / 1 = -3.
Step 2: Determine the slopes of the altitudes.
- The slope of the altitude from A to line BC is the negative reciprocal of the slope of line BC:
slope_altitude_A = -1 / slope_BC = -1 / 1 = -1.
- The slope of the altitude from B to line AC is the negative reciprocal of the slope of line AC:
slope_altitude_B = -1 / slope_AC = -1 / -3 = 1/3.
- The slope of the altitude from C to line AB is the negative reciprocal of the slope of line AB:
slope_altitude_C = -1 / slope_AB = -1 / 0 (undefined).
Since slope_AB is 0, the equation of line AB is simply x = 4.
Step 3: Determine the equations of the altitudes.
- The equation of the altitude from A to line BC can be determined using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line:
y - 6 = -1(x - 1).
y - 6 = -x + 1.
y = -x + 7.
- The equation of the altitude from B to line AC using the same method:
y - 6 = 1/3(x - 4).
y - 6 = 1/3x - 4/3.
y = 1/3x - 4/3 + 18/3.
y = 1/3x + 14/3.
- The equation of the altitude from C to line AB is simply x = 1.
Step 4: Find the intersection points of the altitudes.
To find the orthocenter, we need to find the point where all three altitudes intersect. Solving any two altitude equations will give us one intersection point, and then we can find the third intersection.
- Solving (1) and (2):
From equation (1): y = -x + 7.
Substituting this in equation (2): 1/3x + 14/3 = -x + 7.
1/3x + x = 7 - 14/3.
(1/3+3/3)x = 21/3 - 14/3.
(4/3)x = 7/3.
x = 7/3 * 3/4 = 7/4.
Substituting x in equation (1): y = -(7/4) + 7 = 21/4 - 28/4 = -7/4.
Therefore, the first intersection point is (7/4, -7/4).
- Solving (1) and (3):
From equation (1): y = -x + 7.
Substituting this in equation (3): x = 1.
Therefore, the second intersection point is (1, -6).
- Solving (2) and (3):
From equation (2): y = 1/3x + 14/3.
Substituting this in equation (3): x = 1.
Therefore, the third intersection point is (1, 1).
Now, let's sketch the graph of triangle ABC and mark the intersection point of the altitudes as the orthocenter (H).
(2,5)---(1,3)
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| / |
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(0,3)---(4,3)
From the sketch, we can see that the orthocenter H is the point (1, 3).
Therefore, the steps to find the orthocenter involved finding the slopes of the sides of the triangle, determining the slopes of the altitudes, finding the equations of the altitudes, and solving the intersections of the altitudes to find the orthocenter point.
i dont understand can you clarify??
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
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.