The vertices of a triangle are (0,3) ,(-3,0) and (3,0) . The coordinates of its orthocentre are

Options

A. (0,2)
B. (0,-3)
C. (0,3)
D. (0,-2)

Answer

To find the orthocenter of the triangle with vertices at \( A(0, 3) \), \( B(-3, 0) \), and \( C(3, 0) \), we can follow these steps:

### Step 1: Plot the Points
First, we plot the points \( A(0, 3) \), \( B(-3, 0) \), and \( C(3, 0) \) on the coordinate plane.

### Step 2: Identify the Slope of Line AB
Next, we find the slope of line \( AB \) using the formula for the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A(0, 3) \) and \( B(-3, 0) \):
\[ m_{AB} = \frac{0 - 3}{-3 - 0} = \frac{-3}{-3} = 1 \]

### Step 3: Find the Angle of Line AB
The slope \( m = 1 \) corresponds to an angle \( \theta \) where:
\[ \tan \theta = 1 \implies \theta = 45^\circ \]

### Step 4: Identify the Slope of Line AC
Now, we find the slope of line \( AC \) using the same formula:
\[ m_{AC} = \frac{0 - 3}{3 - 0} = \frac{-3}{3} = -1 \]

### Step 5: Find the Angle of Line AC
The slope \( m = -1 \) corresponds to an angle \( \phi \) where:
\[ \tan \phi = -1 \implies \phi = 135^\circ \]

### Step 6: Determine the Angles of Triangle ABC
Since triangle \( ABC \) is formed, we can find the angles:
- Angle \( A \) is \( 90^\circ \) (as it is formed by the intersection of lines with slopes 1 and -1).
- Angles \( B \) and \( C \) are both \( 45^\circ \).

### Step 7: Identify the Orthocenter
In a right triangle, the orthocenter is located at the vertex where the right angle is formed. Therefore, the orthocenter of triangle \( ABC \) is at point \( A(0, 3) \).

### Final Answer
Thus, the coordinates of the orthocenter are \( (0, 3) \).

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