https://file.aiquickdraw.com/mathbot/file/1727440843098xd2b840f.jpg

The problem in the image is written in Persian and can be translated as follows:

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**Problem 7**:  
The vertices of the triangle are \( A(2,3), B(k,1), C(-1,0) \).  
Calculate the slopes of the sides and determine if there is a value of \( k \) such that \( AB \perp BC \) (i.e., the angle at vertex \( B \) is \( 90^\circ \)).

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### To solve this:

1. **Find the slope of side \( AB \)**:
   - The slope of a line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
     \[
     \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1}
     \]
     Here, \( A(2, 3) \) and \( B(k, 1) \), so:
     \[
     \text{slope of } AB = \frac{1 - 3}{k - 2} = \frac{-2}{k - 2}
     \]

2. **Find the slope of side \( BC \)**:
   - Using the same formula, for \( B(k,1) \) and \( C(-1,0) \):
     \[
     \text{slope of } BC = \frac{0 - 1}{-1 - k} = \frac{-1}{-1 - k} = \frac{1}{1+k}
     \]

3. **Condition for perpendicularity**:
   - Two lines are perpendicular if the product of their slopes is \( -1 \). Thus, we set the product of the slopes of \( AB \) and \( BC \) equal to \( -1 \):
     \[
     \left(\frac{-2}{k - 2}\right) \times \left(\frac{1}{1 + k}\right) = -1
     \]
     Simplifying the equation:
     \[
     \frac{-2}{(k - 2)(1 + k)} = -1
     \]
     Multiplying both sides by \( (k - 2)(1 + k) \):
     \[
     -2 = -(k - 2)(1 + k)
     \]
     Expanding the right-hand side:
     \[
     -2 = -(k^2 + k - 2k - 2) = -(k^2 - k - 2)
     \]
     Thus:
     \[
     -2 = -k^2 + k + 2
     \]
     Simplifying:
     \[
     k^2 - k - 4 = 0
     \]
     This is a quadratic equation.

4. **Solve the quadratic equation**:
   The equation is:
   \[
   k^2 - k - 4 = 0
   \]
   Using the quadratic formula:
   \[
   k = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)} = \frac{1 \pm \sqrt{1 + 16}}{2} = \frac{1 \pm \sqrt{17}}{2}
   \]
   So, the solutions for \( k \) are:
   \[
   k = \frac{1 + \sqrt{17}}{2} \quad \text{or} \quad k = \frac{1 - \sqrt{17}}{2}
   \]

These are the possible values of \( k \) that make \( AB \perp BC \).

Would you like more details or have any further questions?

### Related Questions:
1. How do we derive the quadratic formula?
2. What does it mean for two lines to be perpendicular in terms of their slopes?
3. How can we verify the solution geometrically?
4. How does the Pythagorean theorem apply to this triangle?
5. How do we interpret the discriminant \( \sqrt{17} \) in this context?

### Tip:
Always check your slope calculation carefully, especially when dealing with perpendicular lines, as it's crucial for ensuring accuracy!

The vertices of a triangle are A(2,3), B(k,1), C(-1,0). Calculate the slopes of the sides and determine if there is a value of k such that AB is perpendicular to BC (the angle at vertex B is 90 degrees).

In this problem, we explore how to calculate the slopes of sides AB and BC for a triangle with vertices A(2,3), B(k,1), and C(-1,0). We solve for the value of k that makes AB perpendicular to BC, including solving a quadratic equation. Suitable for Grades 9-12.

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