Lecture on Problems in Coordinate Geometry

Key Concepts

• Vertical and horizontal lines
• Area of a right triangle
• Midpoint formula
• Distance formula
• Equation of a circle
• Tangent line
• Intercepts and graphing
• Distance in 3D
• Area of shaded regions
• Equilateral triangle
• Median of a triangle
• Perpendicular bisector
• Altitude of a triangle

Detailed Notes

Finding the Coordinates of Point B

• Vertical Line: x-coordinates are the same.
• Horizontal Line: y-coordinates are the same.
  • Example: B shares x-coordinate with A (1), and y-coordinate with C (2).

Area of Right Triangle

• Formula: (\text{Area} = \frac{1}{2} \times \text{base} \times \text{height})
• Base: Distance between B and C is 4 units.
• Height: Distance between A and B is 3 units.
• Area: (\frac{1}{2} \times 4 \times 3 = 6) square units.

Center of a Circle

• Endpoints of diameter: (1, 2) and (7, 10).
• Midpoint Formula: (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
  • Center: (4, 6)

Area and Circumference of Circle

• Radius: Use distance formula between center and endpoint.
  • Distance: 5
• Area: (\pi r^2 = 25\pi)
• Circumference: (2\pi r = 10\pi)

Standard Equation of Circle

• Equation: ((x - h)^2 + (y - k)^2 = r^2)
  • Center: (4, 6), Radius: 5
  • Equation: ((x - 4)^2 + (y - 6)^2 = 25)

Equation of Tangent Line

• Slope: Perpendicular slope is negative reciprocal.
• Example, given points A (3, 2) and B (5, 8):
  • Slope of AB: 3
  • Slope of Tangent: -1/3
  • Equation: ( y = -\frac{1}{3}x + \frac{29}{3})

Area of Region Bounded by Axes

• Equation: (4x + y = 8)
• Intercepts:
  • x-intercept: (2, 0)
  • y-intercept: (0, 8)
• Area: (\frac{1}{2} \times 2 \times 8 = 8) square units

Distance in 3D

• Points: Origin (0, 0, 0) and P (3, 4, 5)
• Distance Formula: (\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2})
  • Distance: 5√2 ≈ 7.07

Distance from Point to Line

• Line: (3x + 4y - 7 = 0)
• Point: (5, 3)
• Distance Formula: (\frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}})
  • Distance: 4 units

Area of a Shaded Region

• Circle with x-intercept 4,0 and y-intercept 0,4
• Area of Square: Side length = 8
• Area of Circle: (\pi (4)^2 = 16\pi)
• Shaded Area: 64 - 16\pi

Equilateral Triangle

• Side length: 8
• Area: (16\sqrt{3}) square units

Median of a Triangle

• Points: A (1, 2), B (5, 10), C (7, 4)
• Midpoint of AC: (4, 3)
• Equation of Median: y = 7x - 25

Perpendicular Bisector

• Bisector Equation: Passes through midpoint, slope is negative reciprocal

Altitude of a Triangle

• Passes through vertex, perpendicular to base
• Equation: Shares same slope with perpendicular bisector, different point
• Altitude Equation: y = -3x + 25