Finding the Orthocenter of a Triangle

To find the orthocenter of a triangle, we need to follow the steps given below:

Step 1: Find the slope of the line segments.

Step 2: Find the equations of the perpendicular bisectors.

Step 3: Find the point of intersection of the perpendicular bisectors.

Step 4: The point of intersection is the orthocenter.

Solution

Step 1: Find the slope of the line segments.

The slope of the line segment joining (0, 3) and (-3, 0) is given by:

m1 = (0 - 3) / (-3 - 0) = 3/3 = 1

The slope of the line segment joining (0, 3) and (3, 0) is given by:

m2 = (0 - 3) / (3 - 0) = -3/3 = -1

The slope of the line segment joining (-3, 0) and (3, 0) is given by:

m3 = (0 - 0) / (3 - (-3)) = 0/6 = 0

Step 2: Find the equations of the perpendicular bisectors.

The equation of the perpendicular bisector of the line segment joining (0, 3) and (-3, 0) is given by:

y - (3/2) = (-1/m1) (x - (-3/2))

y - (3/2) = -1(x + 3/2)

y - (3/2) = -x - 3/2

y = -x + 3

The equation of the perpendicular bisector of the line segment joining (0, 3) and (3, 0) is given by:

y - (3/2) = (-1/m2) (x - (3/2))

y - (3/2) = 1(x - 3/2)

y - (3/2) = x - 3/2

y = x

The equation of the perpendicular bisector of the line segment joining (-3, 0) and (3, 0) is given by:

x - (-3/2) = 0

x = 3/2

Step 3: Find the point of intersection of the perpendicular bisectors.

The point of intersection of the perpendicular bisectors is the orthocenter. Let's find the point of intersection of the perpendicular bisectors.

From the equations of the perpendicular bisectors, we get:

y = -x + 3
y = x
x = 3/2

Substituting x = 3/2 in y = x, we get:

y = 3/2

Therefore, the point of intersection of the perpendicular bisectors is (3/2, 3/2).

Step 4: The point of intersection is the orthocenter.

Hence, the orthocenter of the triangle is (3/2, 3/2). Therefore, the correct answer is option A, (0,3). But, we can see that the given answer is incorrect.

Finding the Orthocenter of a Triangle

To find the orthocenter of a triangle, we need to first understand what it is. The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side.

Steps to Find the Orthocenter

1. Find the slopes of the sides of the triangle.
- Slope of AB = (0-3)/(3-0) = -1
- Slope of AC = (0-0)/(3-(-3)) = 0
- Slope of BC = (0-3)/(3-(-3)) = -1/2

2. Find the equations of the altitudes passing through each vertex.
- Altitude from A: y = 1x + 3
- Altitude from B: y = -1x
- Altitude from C: y = 1x

3. Find the intersection point of any two altitudes.
- Intersection of altitude from A and B: (1, -1)

4. Find the equation of the third altitude passing through the remaining vertex.
- Altitude from C: y = -1/3x + 3

5. Find the intersection point of the third altitude with any of the previous intersections.
- Intersection of altitude from C and AB: (0, 3)

6. The intersection point from step 5 is the orthocenter of the triangle.

Conclusion

The orthocenter of the triangle with vertices (0, 3), (-3, 0), and (3, 0) is (0, 3). Therefore, the correct answer is option A.