Finding the Orthocenter of Triangle ABC
To find the orthocenter of triangle ABC with vertices A(0, 6), B(4, 6), and C(1, 3), we first need to understand what the orthocenter is. The orthocenter is the point where the three altitudes of a triangle intersect.
Step 1: Sketch the Triangle
Let’s plot the points:
- A: (0, 6)
- B: (4, 6)
- C: (1, 3)
On a graph, you will see A and B lying on the same horizontal line (y = 6) while point C is below this line.
To sketch the triangle, connect points A, B, and C:
Step 2: Find the Slopes
The first step in finding the altitudes is to find the slopes of the sides of the triangle:
- Line AB: Points A (0,6) and B (4,6) have a slope of 0 (horizontal).
- Line AC: Slope = (y2 – y1) / (x2 – x1) = (3 – 6) / (1 – 0) = -3.
- Line BC: Slope = (3 – 6) / (1 – 4) = 1.
Step 3: Finding the Altitudes
Next, we need to determine the equations of the altitudes:
- Altitude from C to AB: Since AB is horizontal, this altitude will be a vertical line passing through C (1, 3), so it has the equation
x = 1. - Altitude from A to BC: The slope of BC is 1, thus the slope of the altitude from A will be
-1(the negative reciprocal). Using point-slope form:
y - 6 = -1(x - 0) => y = -x + 6 - Altitude from B to AC: The slope of AC is -3, hence the altitude’s slope will be
1/3. Again, using point-slope form:
y - 6 = (1/3)(x - 4) => y = (1/3)x + 20/3
Step 4: Find the Intersection of Two Altitudes
To locate the orthocenter, we can find the intersection of any two altitudes. Let’s find the intersection of the altitudes from A and C:
- From altitude from C:
x = 1 - Substituting
x = 1into the equation of altitude from A:y = -1 + 6 = 5
Thus, we have the point (1, 5).
Conclusion
The orthocenter of triangle ABC is found at the coordinates (1, 5).