We are given three points in a 2D plane: A(1,3), B(-2,-3), and C(3,1), forming a triangle. The problem is asking to find the equation of the line that represents the altitude from point B, which is perpendicular to side AC.

Steps to Solve:

1. Find the slope of line AC: The slope $m_{AC}$ of line passing through two points A(1,3) and C(3,1) is given by:

   $m_{AC} = \frac{y_C - y_A}{x_C - x_A} = \frac{1 - 3}{3 - 1} = \frac{-2}{2} = -1$

2. Find the slope of the altitude (perpendicular to AC): The slope of the altitude (denoted as $m_{\text{alt}}$) will be the negative reciprocal of the slope of AC because the altitude is perpendicular to line AC. Thus,

   $m_{\text{alt}} = -\frac{1}{m_{AC}} = -\frac{1}{-1} = 1$

3. Find the equation of the altitude: The equation of a line is given by:

   $y - y_1 = m(x - x_1)$

   where $m$ is the slope of the line and $(x_1, y_1)$ is a point on the line. In this case, the line passes through point B(-2, -3) and has a slope $m_{\text{alt}} = 1$.

   Substituting the values:

   $y - (-3) = 1(x - (-2))$

   Simplifying:

   $y + 3 = x + 2$

   $y = x - 1$

Thus, the equation of the altitude from point B is:

$y = x - 1$

Let me know if you would like any additional details or clarifications!

Related Questions:

1. How do you calculate the equation of a line given two points?
2. What is the geometric significance of an altitude in a triangle?
3. How can you find the intersection of two lines in 2D geometry?
4. How do perpendicular lines relate in terms of their slopes?
5. Can you derive the equation of the line passing through two specific points?

Tip:

Remember that the product of the slopes of two perpendicular lines in 2D geometry is always -1.