Determine whether the triangle formed by the lines `x-7y+12=0,7x+y-16=0` and `3x+4y-4=0` is

Options

A. equilateral
B. right angled
C. isosceles
D. none

Answer

To determine whether the triangle formed by the lines \( x - 7y + 12 = 0 \), \( 7x + y - 16 = 0 \), and \( 3x + 4y - 4 = 0 \) is equilateral, right-angled, or unknown, we will follow these steps:

### Step 1: Find the slopes of the lines

1. **Convert each line to slope-intercept form (y = mx + b)**:
- For \( L_1: x - 7y + 12 = 0 \):
\[ 7y = x + 12 \implies y = \frac{1}{7}x + \frac{12}{7} \]
Thus, the slope \( m_1 = \frac{1}{7} \).

- For \( L_2: 7x + y - 16 = 0 \):
\[ y = -7x + 16 \]
Thus, the slope \( m_2 = -7 \).

- For \( L_3: 3x + 4y - 4 = 0 \):
\[ 4y = -3x + 4 \implies y = -\frac{3}{4}x + 1 \]
Thus, the slope \( m_3 = -\frac{3}{4} \).

### Step 2: Check if any two lines are perpendicular

2. **Check the product of the slopes**:
- For \( L_1 \) and \( L_2 \):
\[ m_1 \cdot m_2 = \frac{1}{7} \cdot (-7) = -1 \]
Since the product is -1, lines \( L_1 \) and \( L_2 \) are perpendicular.

### Step 3: Determine the angles between the lines

3. **Check if the triangle is right-angled**:
- Since \( L_1 \) and \( L_2 \) are perpendicular, we can conclude that the triangle formed by these lines is a right-angled triangle.

### Step 4: Check for equilateral properties

4. **Calculate the lengths of the sides of the triangle**:
- To find the vertices of the triangle, we need to find the intersection points of the lines:
- **Intersection of \( L_1 \) and \( L_2 \)**:
\[ x - 7y + 12 = 0 \quad \text{and} \quad 7x + y - 16 = 0 \]
Solving these equations simultaneously will give the coordinates of one vertex.

- **Intersection of \( L_2 \) and \( L_3 \)**:
\[ 7x + y - 16 = 0 \quad \text{and} \quad 3x + 4y - 4 = 0 \]
Similarly, solve these equations for another vertex.

- **Intersection of \( L_1 \) and \( L_3 \)**:
\[ x - 7y + 12 = 0 \quad \text{and} \quad 3x + 4y - 4 = 0 \]
Solve for the third vertex.

5. **Calculate the lengths of the sides** using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
After calculating the lengths of all three sides, check if all sides are equal for an equilateral triangle.

### Conclusion

- If two sides are equal, the triangle is isosceles.
- If all three sides are equal, the triangle is equilateral.
- If one angle is \( 90^\circ \), the triangle is right-angled.