The two line segment joining `(-2,7), (-5,-3) and (-8,-13),(1,17)` cut each other at

Options

A. only one point
B. no point
C. infinite number of points
D. none of these

Answer

To find the point of intersection of the two line segments joining the points (-2, 7) to (-5, -3) and (-8, -13) to (1, 17), we will derive the equations of both lines and analyze their relationship.

### Step 1: Find the equation of the first line segment

The first line segment connects the points A(-2, 7) and B(-5, -3).

Using the two-point form of the line equation:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]
Where \((x_1, y_1) = (-2, 7)\) and \((x_2, y_2) = (-5, -3)\).

Substituting the values:
\[ \frac{y - 7}{-3 - 7} = \frac{x + 2}{-5 + 2} \]
This simplifies to:
\[ \frac{y - 7}{-10} = \frac{x + 2}{-3} \]

Cross-multiplying gives:
\[ 3(y - 7) = -10(x + 2) \]
Expanding both sides:
\[ 3y - 21 = -10x - 20 \]
Rearranging the equation:
\[ 10x + 3y - 1 = 0 \quad \text{(Equation 1)} \]

### Step 2: Find the equation of the second line segment

The second line segment connects the points C(-8, -13) and D(1, 17).

Using the same two-point form:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]
Where \((x_1, y_1) = (-8, -13)\) and \((x_2, y_2) = (1, 17)\).

Substituting the values:
\[ \frac{y + 13}{17 + 13} = \frac{x + 8}{1 + 8} \]
This simplifies to:
\[ \frac{y + 13}{30} = \frac{x + 8}{9} \]

Cross-multiplying gives:
\[ 9(y + 13) = 30(x + 8) \]
Expanding both sides:
\[ 9y + 117 = 30x + 240 \]
Rearranging the equation:
\[ 30x - 9y + 123 = 0 \quad \text{(Equation 2)} \]

### Step 3: Analyze the equations

Now we have two equations:
1. \(10x + 3y - 1 = 0\)
2. \(30x - 9y + 123 = 0\)

To check if these lines are the same, we can manipulate Equation 1:
Multiplying Equation 1 by 3:
\[ 30x + 9y - 3 = 0 \]

Now, if we compare this with Equation 2:
\[ 30x - 9y + 123 = 0 \]

We can see that:
- The coefficients of \(x\) are the same.
- The coefficients of \(y\) are opposite.
- The constant terms differ.

This indicates that the two lines are not the same but are parallel.

### Step 4: Conclusion

Since the two lines are parallel, they do not intersect at any point. Therefore, the answer to the question is that the two line segments do not cut each other at any point.