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The problem provides three points \( A(2, 2) \), \( B(-5, -2) \), and \( C(-5, 6) \) in a plane, and asks to find:

a) The length of the line segment \( BC \).  
b) The slope of the line \( AC \).  
c) The area of the triangle \( ABC \).

### Let's go step by step:

#### a) Length of \( BC \):
The distance between two points \( B(x_1, y_1) \) and \( C(x_2, y_2) \) is given by the formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Using \( B(-5, -2) \) and \( C(-5, 6) \), we substitute the coordinates into the formula.

#### b) Slope of the line \( AC \):
The slope of a line passing through two points \( A(x_1, y_1) \) and \( C(x_2, y_2) \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using \( A(2, 2) \) and \( C(-5, 6) \), we substitute the coordinates into the slope formula.

#### c) Area of triangle \( ABC \):
The area of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by the formula:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
We will substitute the coordinates \( A(2, 2) \), \( B(-5, -2) \), and \( C(-5, 6) \) into this formula.

I'll now calculate each part.Here are the results for each part:

a) The length of the line segment \( BC \) is \( 8 \) units.

b) The slope of the line \( AC \) is approximately \( -0.571 \).

c) The area of the triangle \( ABC \) is \( 28 \) square units.

Would you like further details or explanations? Here are some related questions to expand on this:

1. How can you calculate the midpoint of line segment \( BC \)?
2. What is the equation of the line passing through points \( A \) and \( C \)?
3. Can the coordinates of the centroid of triangle \( ABC \) be found?
4. How does the slope relate to perpendicularity or parallelism of lines?
5. What is the perimeter of triangle \( ABC \)?

**Tip:** For verifying geometric solutions, always cross-check distances and areas using multiple methods when possible.

In a plane, three points A(2, 2), B(-5, -2), and C(-5, 6) are given. Solve the following: a) Find the length of BC, b) Find the slope of AC, c) Calculate the area of triangle ABC.

Learn how to solve geometry problems involving three points in a plane. Calculate the length of line BC, the slope of line AC, and the area of triangle ABC with detailed formulas and solutions. This is suitable for students in Grades 9-11.

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