ABC has vertices A(0, 6), B(4, 6), and C(1, 3). Find the orthocenter of ABC?
the orthocenter is where the 3 altitudes of the triangle intersect
find the equation of an altitude by using point-slope
... the point is the vertex (A, B, or C)
... the slope is the perpendicular to the opposite side (negative-reciprocal)
The orthocenter is (1,5)
To find the orthocenter of triangle ABC, we need to find the intersection point of the altitudes of the triangle.
Step 1: Find the slopes of the sides of the triangle.
The slope of side AB is (change in y / change in x) = (6 - 6) / (4 - 0) = 0 / 4 = 0.
The slope of side AC is (change in y / change in x) = (3 - 6) / (1 - 0) = -3 / 1 = -3.
The slope of side BC is (change in y / change in x) = (3 - 6) / (1 - 4) = -3 / -3 = 1.
Step 2: Find the slopes of the altitudes.
The slope of the altitude from vertex A will be the negative reciprocal of the slope of side BC.
So the slope of the altitude from A is -1/1 = -1.
The slope of the altitude from vertex B will be the negative reciprocal of the slope of side AC.
So the slope of the altitude from B is -1/-3 = 1/3.
The slope of the altitude from vertex C will be the negative reciprocal of the slope of side AB.
Since the slope of side AB is 0, the slope of the altitude from C will be undefined.
Step 3: Find the equations of the altitudes.
To find the equation of a line, we need a point on the line and the slope of the line.
Using point-slope form, the equation of the altitude from A is y - 6 = -1(x - 0) -> y - 6 = -x.
Using point-slope form, the equation of the altitude from B is y - 6 = (1/3)(x - 4) -> y - 6 = (1/3)x - (4/3).
Since the slope of the altitude from C is undefined, the equation is simply x = 1 since the line is vertical.
Step 4: Find the intersection point of the altitudes.
To find the orthocenter, we need to find the point where the three altitudes intersect. This point will be the orthocenter.
To find the intersection point, we need to solve the system of equations:
y - 6 = -x,
y - 6 = (1/3)x - (4/3),
x = 1.
We can substitute the value of x = 1 into the second equation to solve for y:
y - 6 = (1/3)(1) - (4/3) -> y - 6 = 1/3 - 4/3 -> y - 6 = -3/3 -> y - 6 = -1 -> y = 5.
Therefore, the orthocenter of triangle ABC is the point (1, 5).
To find the orthocenter of triangle ABC, follow these steps:
Step 1: Find the slopes of AB and AC.
- The slope of AB is (change in y)/(change in x) = (6-6)/(4-0) = 0
- The slope of AC is (change in y)/(change in x) = (3-6)/(1-0) = -3
Step 2: Find the perpendicular slopes of AB and AC.
- The perpendicular slope of AB is the negative reciprocal of its slope. Since the slope is 0, the perpendicular slope is undefined.
- The perpendicular slope of AC is the negative reciprocal of its slope. The negative reciprocal of -3 is 1/3.
Step 3: Find the equations of the altitudes from vertices A, B, and C.
- The altitude from vertex A is perpendicular to side BC and passes through point A(0, 6). Since the perpendicular slope is undefined, the equation of the altitude is x = 0.
- The altitude from vertex B is perpendicular to side AC and passes through point B(4, 6). The equation of the altitude can be found using the point-slope form: (y - y1) = m(x - x1). Plugging in the values, we get (y - 6) = (1/3)(x - 4), which simplifies to y - 6 = (1/3)x - 4/3.
- The altitude from vertex C is perpendicular to side AB and passes through point C(1, 3). The equation of the altitude can be found using the point-slope form: (y - y1) = m(x - x1). Plugging in the values, we get (y - 3) = (0)(x - 1), which simplifies to y - 3 = 0.
Step 4: Solve the system of equations formed by the altitudes.
- Since x = 0 is the equation of altitude from vertex A, we can substitute this value into the other two altitude equations.
- Substituting x = 0 into the equation of the altitude from vertex B, we get y - 6 = (1/3)(0 - 4), which simplifies to y - 6 = -4/3. Solving for y, we get y = 14/3.
- Substituting x = 0 into the equation of the altitude from vertex C, we get y - 3 = 0. Solving for y, we get y = 3.
Step 5: Find the coordinates of the orthocenter.
- The orthocenter is the point where the three altitudes intersect. Since x = 0 is the equation of the altitude from vertex A and y = 3 is the equation of the altitude from vertex C, the orthocenter must have coordinates (0, 3).
Therefore, the orthocenter of triangle ABC is (0, 3).