The triangle formed by the straight lines `x=y`, `x+y=4` and `x+3y=4` is :

Options

A. isosceles
B. equilateral
C. right-angled
D. None of these

Answer

To determine the nature of the triangle formed by the lines \(x = y\), \(x + y = 4\), and \(x + 3y = 4\), we will follow these steps:

### Step 1: Find the points of intersection of the lines

1. **Intersection of \(x = y\) and \(x + y = 4\)**:
- Substitute \(y\) from \(x = y\) into \(x + y = 4\):
\[ x + x = 4 \implies 2x = 4 \implies x = 2 \implies y = 2 \]
- So, the intersection point is \((2, 2)\).

2. **Intersection of \(x = y\) and \(x + 3y = 4\)**:
- Substitute \(y\) from \(x = y\) into \(x + 3y = 4\):
\[ x + 3x = 4 \implies 4x = 4 \implies x = 1 \implies y = 1 \]
- So, the intersection point is \((1, 1)\).

3. **Intersection of \(x + y = 4\) and \(x + 3y = 4\)**:
- Subtract the first equation from the second:
\[ (x + 3y) - (x + y) = 4 - 4 \implies 2y = 0 \implies y = 0 \]
- Substitute \(y = 0\) into \(x + y = 4\):
\[ x + 0 = 4 \implies x = 4 \]
- So, the intersection point is \((4, 0)\).

### Step 2: Identify the vertices of the triangle

The vertices of the triangle formed by the lines are:
- \(A(2, 2)\)
- \(B(1, 1)\)
- \(C(4, 0)\)

### Step 3: Calculate the lengths of the sides of the triangle

1. **Length of side \(AB\)**:
\[ AB = \sqrt{(2 - 1)^2 + (2 - 1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

2. **Length of side \(BC\)**:
\[ BC = \sqrt{(4 - 1)^2 + (0 - 1)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

3. **Length of side \(CA\)**:
\[ CA = \sqrt{(4 - 2)^2 + (0 - 2)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 4: Determine the nature of the triangle

To determine if the triangle is right-angled, we can check the slopes of the lines:

1. **Slope of line \(x = y\)**:
- This line has a slope of \(1\).

2. **Slope of line \(x + y = 4\)**:
- Rearranging gives \(y = -x + 4\), so the slope is \(-1\).

3. **Slope of line \(x + 3y = 4\)**:
- Rearranging gives \(y = -\frac{1}{3}x + \frac{4}{3}\), so the slope is \(-\frac{1}{3}\).

### Step 5: Check for perpendicular lines

- The product of the slopes of \(x = y\) (slope = 1) and \(x + y = 4\) (slope = -1) is:
\[ 1 \cdot (-1) = -1 \]
This indicates that these two lines are perpendicular.

### Conclusion

Since one angle of the triangle formed by the lines is \(90^\circ\), the triangle is a right-angled triangle.

### Final Answer
The triangle formed by the lines \(x = y\), \(x + y = 4\), and \(x + 3y = 4\) is a right-angled triangle.

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