The orthocentre of the triangle with vertices `(0,0),(3,4),` and `(4,0)` is

Options

A. `(3,5/4)`
B. `(3,12)`
C. `(3,3/4)`
D. `(3,9)`

Answer

To find the orthocenter of the triangle with vertices \( A(0,0) \), \( B(3,4) \), and \( C(4,0) \), we will follow these steps:

### Step 1: Find the slopes of the sides of the triangle
1. **Calculate the slope of line segment AB**:
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{3 - 0} = \frac{4}{3} \]

2. **Calculate the slope of line segment BC**:
\[ \text{slope of } BC = \frac{0 - 4}{4 - 3} = \frac{-4}{1} = -4 \]

3. **Calculate the slope of line segment AC**:
\[ \text{slope of } AC = \frac{0 - 0}{4 - 0} = 0 \]

### Step 2: Find the equations of the altitudes
1. **Find the equation of the altitude from vertex A (0,0) to line BC**:
- The slope of BC is -4, so the slope of the altitude from A will be the negative reciprocal:
\[ \text{slope of altitude from A} = \frac{1}{4} \]
- The equation of the line through A(0,0) with this slope is:
\[ y = \frac{1}{4}x \]

2. **Find the equation of the altitude from vertex B (3,4) to line AC**:
- The slope of AC is 0 (horizontal line), so the altitude from B will be a vertical line:
\[ x = 3 \]

### Step 3: Find the intersection of the altitudes
- Substitute \( x = 3 \) into the equation of the altitude from A:
\[ y = \frac{1}{4}(3) = \frac{3}{4} \]
- Thus, the coordinates of the orthocenter H are:
\[ H(3, \frac{3}{4}) \]

### Final Answer:
The orthocenter of the triangle with vertices \( (0,0) \), \( (3,4) \), and \( (4,0) \) is:
\[ \boxed{\left(3, \frac{3}{4}\right)} \]