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Geometric Shapes: Special Triangles

Jul 17, 2024

Geometric Shapes: Special Triangles Lecture

30-60-90 Triangles

  • Definition: A right-angle triangle with angles of 30, 60, and 90 degrees.
  • Properties:
    • Ratio of sides: 1 : √3 : 2.
    • If one side is known, the others can be calculated using the ratios.
  • Example Calculation:
    • Assume side opposite 30° = 1.
    • Apply tan(60) = √3 to determine perpendicular (P) and base (B).
    • Solve for hypotenuse (H) using Pythagorean theorem; H = 2.

45-45-90 Triangles

  • Definition: A right-angle isosceles triangle with angles of 45, 45, and 90 degrees.
  • Properties:
    • Ratio of sides: 1 : 1 : √2.
    • Equal sides opposite 45° angles.
  • Example Calculation:
    • Assume sides opposite 45° = 1.
    • Hypotenuse becomes √2 using Pythagorean theorem.

Applications of Triangles

  • Height and Distance Problems:
    • Example: A building with a height of 100m is observed from two points with angles of elevation 30° and 45°.
    • Use trigonometric ratios to find distances and other triangle properties.

Equilateral Triangles

  • Definition: A triangle where all sides and angles are equal.
  • Properties:
    • Each angle is 60°.
    • Incenter, circumcenter, centroid, and orthocenter coincide at the same point.
    • Symmetrical properties ensure various bisectors and medians are also altitudes.
  • Key Points:
    • Height (h) = a √3 / 2.
    • Area = √3 / 4 * a².
    • Circumradius (R) = a / √3.
    • Inradius (r) = a / (2√3) or (1/3) height.

Example Problem: Combined Circle and Triangles

  • Problem: Given an equilateral triangle with side 2 units, and concentric circles as circumcircle and incircle, find the side of a larger equilateral triangle that circumscribes this outer circle.
  • Solution Steps:
    • Circumradius for inner triangle: 2 / √3.
    • Relate this to the inradius of the larger triangle: x / (2√3) = 2 / √3.
    • Simplify to find x = 4.
    • Calculate area of larger triangle: √3 / 4 * 4².

Final Thoughts

  • Application: Understanding these key properties and ratios allows for solving complex geometric problems involving special triangles.
  • Reminder: Watch supplementary trigonometry basics video for more understanding.

Study Further: Review each concept and practice additional problems to solidify understanding.

Full transcript