Lesson 5.8: Coordinate Geometry of Triangles

Learning Objectives

• Calculate side lengths using vertex coordinates.
• Determine slopes of triangle sides and analyze parallel/perpendicular relationships.
• Find midpoints of triangle sides.
• Compute triangle area using coordinate methods.

Topics Covered

• Distance formula for side lengths
• Slope formula and side relationships
• Midpoint formula
• Area of a triangle via coordinates
• Application problems in coordinate geometry

Key Terms

• Distance Formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
• Slope: \(m = \frac{y_2-y_1}{x_2-x_1}\)
• Midpoint: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
• Area of Triangle: \(A = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) |\)

Lesson Examples

Example 1: Calculating Side Lengths

1. Vertices: A(1,2), B(4,6), C(5,3)
2. Side AB: \( \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5\)
3. Side BC: \( \sqrt{(5-4)^2 + (3-6)^2} = \sqrt{1+9} = \sqrt{10}\)
4. Side AC: \( \sqrt{(5-1)^2 + (3-2)^2} = \sqrt{16+1} = \sqrt{17}\)

Example 2: Finding Slopes and Checking Perpendicularity

1. Slope AB: \(m = (6-2)/(4-1) = 4/3\)
2. Slope AC: \(m = (3-2)/(5-1) = 1/4\)
3. Check perpendicularity: \(m_{AB} * m_{AC} = 4/3 * 1/4 = 1/3 \neq -1\) → Not perpendicular

Example 3: Finding Midpoints

1. Midpoint of AB: \(M = ((1+4)/2, (2+6)/2) = (2.5,4)\)
2. Midpoint of BC: \(M = ((4+5)/2, (6+3)/2) = (4.5,4.5)\)

Example 4: Area Using Coordinates

1. Vertices: A(1,2), B(4,6), C(5,3)
2. Area: \(A = \frac{1}{2}|1(6-3) + 4(3-2) + 5(2-6)| = \frac{1}{2}|3 + 4 -20| = \frac{1}{2}*13 = 6.5\)

Practice Problems

1. Vertices: P(2,1), Q(5,4), R(6,2). Find all side lengths and classify the triangle by sides.
2. Check if the triangle with vertices A(0,0), B(3,3), C(3,0) has any perpendicular sides.
3. Find the area of a triangle with vertices (1,1), (4,5), (6,2).
4. Find the midpoints of sides of triangle with vertices (0,0), (2,4), (4,0).

Tips & Common Mistakes

• Always double-check subtraction when computing slopes or distances.
• Area formula requires absolute value to avoid negative areas.
• Perpendicular lines have slopes that multiply to -1.

Summary

Coordinate geometry allows precise computation of triangle properties. Using formulas for distance, slope, midpoint, and area, students can analyze and solve problems for any triangle on the coordinate plane.

Challenge Problems

1. Vertices: A(-1,2), B(3,6), C(5,-2). Compute side lengths, slopes, midpoints, and area.
2. Check if triangle with vertices (0,0), (4,4), (4,0) is a right triangle using slopes.
3. Find the centroid of a triangle with vertices (2,3), (6,7), (4,1).