The orthocentre of the triangle whose vertices are `(1, 1), (5, 1) and (4, 5)` is

Options

A. `((9)/(4), -(1)/(3))`
B. `(4, 13)`
C. `(4, (9)/(4))`
D. `(4, (7)/(4))`

Answer

To find the orthocenter of the triangle with vertices \( A(1, 1) \), \( B(5, 1) \), and \( C(4, 5) \), we will follow these steps:

### Step 1: Find the slopes of the sides of the triangle
1. **Calculate the slope of side \( AB \)**:
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 1}{5 - 1} = 0 \]
The slope of line \( AB \) is \( 0 \) (horizontal line).

2. **Calculate the slope of side \( BC \)**:
\[ \text{slope of } BC = \frac{5 - 1}{4 - 5} = \frac{4}{-1} = -4 \]

3. **Calculate the slope of side \( CA \)**:
\[ \text{slope of } CA = \frac{5 - 1}{4 - 1} = \frac{4}{3} \]

### Step 2: Find the slopes of the altitudes
1. **The slope of the altitude from vertex \( C \) to side \( AB \)**:
Since \( AB \) is horizontal (slope = 0), the altitude from \( C \) will be vertical. Therefore, its slope is undefined (or we can say it is vertical).

2. **The slope of the altitude from vertex \( A \) to side \( BC \)**:
The slope of \( BC \) is \( -4 \). The slope of the altitude from \( A \) will be the negative reciprocal:
\[ \text{slope of altitude from } A = \frac{1}{4} \]

### Step 3: Find the equations of the altitudes
1. **Equation of the altitude from \( C(4, 5) \)**:
Since it is vertical, the equation is:
\[ x = 4 \]

2. **Equation of the altitude from \( A(1, 1) \)**:
Using the point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - 1 = \frac{1}{4}(x - 1) \]
Simplifying:
\[ 4y - 4 = x - 1 \implies x - 4y + 3 = 0 \]

### Step 4: Find the intersection of the two altitudes
To find the orthocenter, we solve the equations:
1. From the vertical line \( x = 4 \).
2. Substitute \( x = 4 \) into the equation \( x - 4y + 3 = 0 \):
\[ 4 - 4y + 3 = 0 \implies 4y = 7 \implies y = \frac{7}{4} \]

### Conclusion
The orthocenter of the triangle is:
\[ \text{Orthocenter} = \left(4, \frac{7}{4}\right) \]

### Final Answer
The orthocenter of the triangle is \( \left(4, \frac{7}{4}\right) \).

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