Geometry and Analytical Geometry

10.1 Introduction

• Geometry, considered a logical system, serves as a powerful means to inspire children about the human spirit. (H. Freudenthal)
• Two-dimensional coordinate geometry is a blend of algebra and geometry.
• René Descartes, a notable French philosopher and mathematician, conducted a systematic study of geometry using algebra, marking the inception of analytical geometry with his work ‘La Géométry’ published in 1637.
    - Introduced concepts such as the equation of a curve.
    - Analytical geometry combines analysis with geometry.
• In previous classes, basic concepts of coordinate geometry were introduced including:
    - Coordinate axes
    - Coordinate plane
    - Plotting points in a plane
    - Distance between points
    - Section formulae

Recall of Coordinate Geometry

• Example of plotting points:
    - Point (6, -4) is located:
      - 6 units from the y-axis (along the positive x-axis)
      - 4 units from the x-axis (along the negative y-axis)
    - Point (3, 0) is located:
      - 3 units from the y-axis (along the positive x-axis)
      - 0 distance from the x-axis.

Important Formulae

• I. Distance between two points P (x1, y1) and Q (x2, y2):
  $PQ = \frac{ ext{√}((x_2-x_1)^2 + (y_2-y_1)^2)}{1}$
     - Example calculation for distance between (6, -4) and (3, 0):
  $= ext{√}(3-6)^2 + (0 + 4)^2 = ext{√}(9 + 16) = ext{√}25 = 5 ext{ units}$

• II. Coordinates of a point dividing a line segment joining points (x1, y1) and (x2, y2) internally in ratio m:n:
  $\frac{m}{n} : (x, y) = \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}$
    - Example for A (1, -3) and B (-3, 9) dividing the line segment in ratio 1:3:
    - This gives:
  $x = \frac{1(-3) + 3(1)}{1+3} = \frac{-3 + 3}{4} = 0$
  $y = \frac{1(9) + 3(-3)}{1+3} = \frac{9 - 9}{4} = 0$

• III. Mid-point of a line segment:
    - If m = n:
  $M = \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}$

• IV. Area of a triangle with vertices (x1, y1), (x2, y2), (x3, y3):
  $ext{Area} = \frac{1}{2} \bigg| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg|$
    - Example for vertices (4, 4), (3, -2), (-3, 16):
  $ext{Area} = \frac{1}{2} \bigg| 4(-2 - 16) + 3(16 - 4) + (-3)(4 + 2) \bigg| = 27$

• Remark: If the area of triangle ABC is zero, then points A, B, and C are collinear.

10.2 Slope of a Line

• A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
• Inclination (θ) of line l:
    - Measures the angle with the positive direction of the x-axis, constrained between 0°≤ θ ≤ 180°.
    - Lines parallel to x-axis have an inclination of 0°; vertical lines have an inclination of 90°.
• Definition of Slope:
    - Slope (m) at inclination θ:
  $m = an(θ), ext{where } θ <br />\neq 90°$
    - Notable Points:
      - Slope of x-axis is zero
      - Slope of y-axis is undefined.

10.2.1 Slope from Coordinates

• Given two points P (x1, y1) and Q (x2, y2) on line l, if $x_1
  eq x_2$:
• Case 1: If θ is acute:
  $m = \frac{y2 - y1}{x2 - x1}$
• Case 2: If θ is obtuse:
  $m = \frac{y2 - y1}{x2 - x1} = - an(∠MPQ)$
    Thus, in both cases, slope is given by:
  $m = \frac{y2 - y1}{x2 - x1}$

10.2.2 Conditions for Parallelism and Perpendicularity

• Non-Vertical lines l1 and l2 with slopes m1 and m2:
• Parallel Lines:
    - If l1 || l2, then,
  $m_1 = m_2$
• Perpendicular Lines:
    - If l1 ⊥ l2, then,
  $m_1 imes m_2 = -1$
• Examples of finding slopes and their conditions given in detail for multiple pairs of points.

10.2.3 Angle Between Two Lines

• When two non-vertical lines with slopes m1 and m2 intersect:
    - Acute angle θ between the lines is given by:
  $an θ = \frac{| m_1 - m_2 |}{1 + m_1 m_2}$
• Obtuse angle φ regarding relationship:
  $φ = 180° - θ$

Example Format:

• Inclined lines examples showcase slope determination, area finding, as well as parallel and perpendicular conditions emphasized through parsing geometrical contexts along with necessary calculations and formula implementations.

10.3 Various Forms of Equation of a Line

• Dedicating sections on understanding the forms of line equations influenced by geometrical attributes such as horizontal, vertical orientation, intercept forms, slope-intercept formats along with general definitions are prominent in well-structured formats showcasing varied scenario applications.

Summary

• Slope (m) of a non-vertical line through points (x1, y1) and (x2, y2):
  $m = \frac{y_2 - y_1}{x_2 - x_1}$ Where $x_1
  eq x_2$.

• Inclined angle (θ) relates:
  $m = an θ$ for line orientation angles.

• Parallelism and perpendicularity utilize slope relationships distinctly in calculations to establish geometric line conditions.

• Area, distance, and coordinate segment relations firmly anchor geometric foundations for further analytical geometry explorations along straight line properties.

• All exercises provided at the end reinforce the comprehensive applications of the slopes, angles, and relationships across lines, incorporating analytical methodologies for successful problem-solving and contextual grasp.