Distance Formula and Polygon Classification Notes

Distance Formula

Definition

• The distance between two points 99dX(dx,y) and (x2,y2) in a Cartesian coordinate system is given by the formula:
  $d = ext{sqrt}((x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2)$

Derived Formula

• The formula can be simplified and rewritten as follows:
  $d = ext{sqrt}( ext{change in x}^2 + ext{change in y}^2)$

Example 1: Calculating Distance

• Problem Statement: Find the exact distance between the points (-10, -2) and (6, -10).

  • Coordinates:
  • Point 1: (x1, y1) = (-10, -2)
  • Point 2: (x2, y2) = (6, -10)

• Solution Steps:

  1. Calculate the difference in x-coordinates:
     $x<em>2 - x</em>1 = 6 - (-10) = 16$
  2. Calculate the difference in y-coordinates:
     $y<em>2 - y</em>1 = -10 - (-2) = -8$
  3. Plug into distance formula:
     $d = ext{sqrt}((16)^2 + (-8)^2)$
  4. Calculate:
     • $d = ext{sqrt}(256 + 64)$
     • $d = ext{sqrt}(320)$
     • $d = 8 ext{sqrt}(5)$

• Answer: The distance is approximately 17.89 units.

Example 2: Finding Missing Coordinate

• Problem Statement: Find the missing coordinate given the following information.

  1. Points (7, -5) and (x, y) where distance d = 4 units.

     • We use:
       $d = ext{sqrt}((x - 7)^2 + (y + 5)^2) = 4$
     • Square both sides:
       $(x - 7)^2 + (y + 5)^2 = 16$

  2. Points (x, -3) and (1, 9) where d = 13 units.

     • We use:
       $d = ext{sqrt}((x - 1)^2 + (-3 - 9)^2) = 13$
     • Square both sides:
       $(x - 1)^2 + 144 = 169$
     • Solve:
       $(x - 1)^2 = 25$
     • Thus,
       x - 1 = ext{±5 99=> x = 6 ext{ or } -4}

  3. The points (6, 4) and (0, y) where d = 56 units.

     • Use:
       $d = ext{sqrt}((0 - 6)^2 + (y - 4)^2) = 56$
     • Square both sides:
       $(36 + (y - 4)^2 = 3136)$
     • Solve:
       $(y - 4)^2 = 3100$
       $y - 4 = ext{± ext{sqrt}}(3100)$

Polygons Classification

Definitions of Polygons

• Rectangle: A quadrilateral with two pairs of parallel sides that are equal in length.
• Square: A special case of rectangle where all four sides are equal and angles are 90 degrees.
• Isosceles Triangle: A triangle with at least two congruent sides.
• Scalene Triangle: A triangle with all sides of different lengths.
• Quadrilateral: A polygon with four sides.

Example 3: Classifying Polygons

• Polygon 1: Points A(-6,0), B(-3,3), C(-6,6), D(-9,3)
  • Classification: Needs to compute side lengths to verify.
• Polygon 2: Points J(-4,-4), U(-11,-4), M(-11,-8), P(-4,-6)
  • Classification: Based on equal side lengths and angles.
• Polygon 3: Points (C(6,4), A(9,7), T(12,4))
  • Classification: Regular polygons need computation for angles and lengths to establish.

Conclusion

• Understanding distance formulae and classifications of geometric shapes is crucial in geometry. The calculations enable determining spatial relationships in a Cartesian coordinate system.
• Practice examples to solidify knowledge of concepts, these principles are foundational in the study of geometry and related fields.