The sides of a triangle are the straight lines `x+y=1,7y=x ,` and `sqrt(3)y+x=0` . Then which of the following is an interior point of the triangle? Circumcenter (b) Centroid Incenter (d) Orthocenter

Options

A. Circumcenter
B. Centroid
C. Incenter
D. Orthocenter

Explanation

To determine which of the given points (circumcenter, centroid, incenter, orthocenter) is an interior point of the triangle formed by the lines \(x + y = 1\), \(7y = x\), and \(\sqrt{3}y + x = 0\), we will follow these steps:

### Step 1: Find the points of intersection of the lines
We need to find the vertices of the triangle by determining the intersections of the three lines.

1. **Intersection of \(x + y = 1\) and \(7y = x\)**:
- Substitute \(x = 7y\) into \(x + y = 1\):
\[ 7y + y = 1 \implies 8y = 1 \implies y = \frac{1}{8} \]
- Now, substituting \(y = \frac{1}{8}\) back to find \(x\):
\[ x = 7 \cdot \frac{1}{8} = \frac{7}{8} \]
- So, the first vertex is \(A\left(\frac{7}{8}, \frac{1}{8}\right)\).

2. **Intersection of \(x + y = 1\) and \(\sqrt{3}y + x = 0\)**:
- Substitute \(x = -\sqrt{3}y\) into \(x + y = 1\):
\[ -\sqrt{3}y + y = 1 \implies (1 - \sqrt{3})y = 1 \implies y = \frac{1}{1 - \sqrt{3}} \]
- Now, substituting \(y\) back to find \(x\):
\[ x = -\sqrt{3} \cdot \frac{1}{1 - \sqrt{3}} = \frac{-\sqrt{3}}{1 - \sqrt{3}} \]
- The second vertex is \(B\left(\frac{-\sqrt{3}}{1 - \sqrt{3}}, \frac{1}{1 - \sqrt{3}}\right)\).

3. **Intersection of \(7y = x\) and \(\sqrt{3}y + x = 0\)**:
- Substitute \(x = 7y\) into \(\sqrt{3}y + x = 0\):
\[ \sqrt{3}y + 7y = 0 \implies (\sqrt{3} + 7)y = 0 \implies y = 0 \]
- Now, substituting \(y = 0\) back to find \(x\):
\[ x = 7 \cdot 0 = 0 \]
- The third vertex is \(C(0, 0)\).

### Step 2: Identify the type of triangle
The vertices of the triangle are \(A\left(\frac{7}{8}, \frac{1}{8}\right)\), \(B\left(\frac{-\sqrt{3}}{1 - \sqrt{3}}, \frac{1}{1 - \sqrt{3}}\right)\), and \(C(0, 0)\).

To determine the type of triangle, we can check the lengths of the sides using the distance formula:
- Length \(AB\), \(AC\), and \(BC\) can be calculated, but we can also analyze the slopes to determine if the triangle is obtuse.

### Step 3: Determine the location of the centers
1. **Circumcenter**: In an obtuse triangle, the circumcenter lies outside the triangle.
2. **Centroid**: The centroid is always inside the triangle.
3. **Incenter**: The incenter is also always inside the triangle.
4. **Orthocenter**: In an obtuse triangle, the orthocenter lies outside the triangle.

### Conclusion
Since the circumcenter and orthocenter lie outside the triangle, and the centroid and incenter lie inside the triangle, the correct options for interior points of the triangle are:
- **Centroid**
- **Incenter**

### Final Answer
The interior points of the triangle are:
- (b) Centroid
- (c) Incenter