Orthocentre of triangle whose vertices are (0,0),(3,4),(4,0) is

Options

A. `(3,7/3)`
B. `(3,5/4)`
C. `(5,-2)`
D. `(3,3/4)`

Answer

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

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are:
- A(0,0)
- B(3,4)
- C(4,0)

### Step 2: Find the slopes of the sides of the triangle
1. **Slope of AB**:
\[ \text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{3 - 0} = \frac{4}{3} \]

2. **Slope of AC**:
\[ \text{slope of AC} = \frac{0 - 0}{4 - 0} = 0 \]

3. **Slope of BC**:
\[ \text{slope of BC} = \frac{0 - 4}{4 - 3} = \frac{-4}{1} = -4 \]

### Step 3: Find the slopes of the altitudes
1. **Altitude from A (perpendicular to BC)**:
The slope of the altitude from A will be the negative reciprocal of the slope of BC:
\[ \text{slope of altitude from A} = \frac{1}{4} \]

2. **Altitude from B (perpendicular to AC)**:
The slope of the altitude from B will be the negative reciprocal of the slope of AC:
\[ \text{slope of altitude from B} = \text{undefined} \quad (\text{since AC is horizontal}) \]
The equation of the altitude from B (vertical line) is:
\[ x = 3 \]

### Step 4: Find the equations of the altitudes
1. **Equation of the altitude from A**:
Using point A(0,0) and slope \(\frac{1}{4}\):
\[ y - 0 = \frac{1}{4}(x - 0) \implies y = \frac{1}{4}x \]

2. **Equation of the altitude from B**:
As derived, the altitude from B is:
\[ x = 3 \]

### Step 5: Find the intersection of the altitudes
To find the orthocenter, we need to solve the equations:
1. \(y = \frac{1}{4}x\)
2. \(x = 3\)

Substituting \(x = 3\) into the first equation:
\[ y = \frac{1}{4}(3) = \frac{3}{4} \]

Thus, the orthocenter is at the point:
\[ (3, \frac{3}{4}) \]

### Final Answer
The orthocenter of the triangle is \((3, \frac{3}{4})\).

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