Understanding the Problem: Foot of the Altitude

The question asks us to find the coordinates of the foot of the altitude drawn from vertex A to the side BC in triangle ABC. The vertices of the triangle are given as A(-2, 3), B(2, 1), and C(1, 2). The foot of the altitude is the point on the line containing side BC where the altitude from A intersects it. The altitude from A is a line segment starting at A and perpendicular to the line containing side BC.

Steps to Find the Foot of the Altitude

To find the foot of the altitude, we need to perform the following steps:

1. Find the equation of the line passing through points B and C (line BC).
2. Find the equation of the altitude line passing through point A and perpendicular to line BC.
3. Find the point of intersection of the line BC and the altitude line. This intersection point is the foot of the altitude.

Step 1: Finding the Equation of Line BC

The line BC passes through points B(2, 1) and C(1, 2). We can find the slope of the line BC using the formula:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

For points B(2, 1) and C(1, 2):

\(m_{BC} = \frac{2 - 1}{1 - 2} = \frac{1}{-1} = -1\)

Now, we can use the point-slope form of the equation of a line \(y - y_1 = m(x - x_1)\) with point B(2, 1) and slope \(m_{BC} = -1\):

\(y - 1 = -1(x - 2)\)

\(y - 1 = -x + 2\)

Adding 1 to both sides, we get the equation of line BC:

\(y = -x + 3\)

Alternatively, we can write it as \(x + y - 3 = 0\).

Step 2: Finding the Equation of the Altitude from A

The altitude from A is perpendicular to the line BC. If two lines are perpendicular, the product of their slopes is -1 (provided neither line is vertical or horizontal). Since the slope of BC is \(m_{BC} = -1\), the slope of the altitude from A (\(m_{alt}\)) is:

\(m_{alt} = -\frac{1}{m_{BC}} = -\frac{1}{-1} = 1\)

The altitude passes through vertex A(-2, 3). Using the point-slope form \(y - y_1 = m(x - x_1)\) with point A(-2, 3) and slope \(m_{alt} = 1\):

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

\(y - 3 = x + 2\)

Adding 3 to both sides, we get the equation of the altitude from A:

\(y = x + 5\)

Alternatively, we can write it as \(x - y + 5 = 0\).

Step 3: Finding the Point of Intersection (Foot of the Altitude)

The foot of the altitude is the point where the line BC and the altitude from A intersect. We need to solve the system of linear equations for these two lines:

Equation 1 (Line BC): \(y = -x + 3\)

Equation 2 (Altitude from A): \(y = x + 5\)

We can use the substitution method. Substitute the expression for \(y\) from Equation 2 into Equation 1:

\(x + 5 = -x + 3\)

Now, solve for \(x\). Add \(x\) to both sides:

\(2x + 5 = 3\)

Subtract 5 from both sides:

\(2x = 3 - 5\)

\(2x = -2\)

Divide by 2:

\(x = -1\)

Now substitute the value of \(x = -1\) back into either Equation 1 or Equation 2 to find \(y\). Using Equation 2:

\(y = x + 5\)

\(y = -1 + 5\)

\(y = 4\)

The point of intersection, which is the foot of the altitude from A, is (-1, 4).

Summary of Calculation Steps

The foot of the altitude from vertex A of the triangle ABC is at the coordinates (-1, 4).

Revision Table: Key Concepts Reviewed

Additional Information: Properties of Altitudes and Orthocenter

Understanding altitudes is important in geometry. Here are some related concepts:

• Every triangle has three altitudes, one from each vertex to the opposite side.
• The three altitudes of a triangle intersect at a single point called the orthocenter.
• For an acute-angled triangle, the orthocenter lies inside the triangle.
• For a right-angled triangle, the orthocenter is at the vertex containing the right angle.
• For an obtuse-angled triangle, the orthocenter lies outside the triangle.
• The coordinates of the orthocenter can be found by finding the intersection of any two altitudes.

This problem focused on finding the foot of a single altitude, which is a fundamental step in understanding triangle properties and finding the orthocenter using coordinate geometry.