Find the coordinates of the orthocentre of the triangle whose vertices are (1,2) ,(2,3) and (4,3) .

Options

A. a. (2,5)
B. b. (3,4)
C. c. (1,6)
D. d. none of these

Answer

To find the coordinates of the orthocenter of the triangle with vertices at \( A(1, 2) \), \( B(2, 3) \), and \( C(4, 3) \), we can follow these steps:

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

2. **Find the slope of side AB**:
\[ \text{slope of AB} = \frac{3 - 2}{2 - 1} = \frac{1}{1} = 1 \]

### Step 2: Determine the slopes of the altitudes
1. **The slope of the altitude from B (perpendicular to AC)**:
Since the product of the slopes of two perpendicular lines is \(-1\):
\[ m_{BD} \cdot m_{AC} = -1 \implies m_{BD} \cdot \frac{1}{3} = -1 \implies m_{BD} = -3 \]

2. **The slope of the altitude from C (perpendicular to AB)**:
\[ m_{CE} \cdot m_{AB} = -1 \implies m_{CE} \cdot 1 = -1 \implies m_{CE} = -1 \]

### Step 3: Write the equations of the altitudes
1. **Equation of altitude BD** (passing through point B(2, 3)):
Using the point-slope form \( y - y_1 = m(x - x_1) \):
\[ y - 3 = -3(x - 2) \implies y - 3 = -3x + 6 \implies 3x + y - 9 = 0 \quad \text{(Equation 1)} \]

2. **Equation of altitude CE** (passing through point C(4, 3)):
\[ y - 3 = -1(x - 4) \implies y - 3 = -x + 4 \implies x + y - 7 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations to find the orthocenter
To find the intersection of the two lines (the orthocenter), we solve the equations:
1. From Equation 1: \( 3x + y - 9 = 0 \)
2. From Equation 2: \( x + y - 7 = 0 \)

**Substituting Equation 2 into Equation 1**:
- From Equation 2, we can express \( y \):
\[ y = 7 - x \]
- Substitute into Equation 1:
\[ 3x + (7 - x) - 9 = 0 \implies 3x - x + 7 - 9 = 0 \implies 2x - 2 = 0 \implies x = 1 \]
- Substitute \( x = 1 \) back into Equation 2 to find \( y \):
\[ y = 7 - 1 = 6 \]

### Conclusion
The coordinates of the orthocenter are \( (1, 6) \).

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