Honors IM3 Unit 6 - Special Rights

  • Learning Focus

    • Objective: Find missing sides of special right triangles without using trigonometry.
    • Types of Special Right Triangles:
    • 30°-60°-90°
    • 45°-45°-90°

  • Pythagorean Theorem

    • Useful for finding missing sides of right triangles.
    • If using Pythagorean theorem: need lengths of two sides (legs and hypotenuse).
    • Right triangle trigonometry requires calculator for values of trigonometric ratios.

  • Special Right Triangles

    • 45°-45°-90° Triangle:
    • Both legs are of equal length: If one leg = x, then hypotenuse = xext√2x ext{√}2.
    • 30°-60°-90° Triangle:
    • If the shortest leg (opposite 30°) = x, then the hypotenuse = 2x2x and the longer leg (opposite 60°) = xext√3x ext{√}3.

  • Generalizations

    • For a 45°-45°-90° triangle:
    • If one side = x, then the other leg = x and hypotenuse = xext√2x ext{√}2.
    • For a 30°-60°-90° triangle:
    • If one side (short leg) = x, then longer leg = xext√3x ext{√}3 and hypotenuse = 2x2x.

  • Checking Understanding

    • Fill in measures for given triangles. Understand trigonometric ratios without calculators.
    • e.g., sin(45°)=cos(45°)=tan(45°)=1sin(45°) = cos(45°) = tan(45°) = 1, sin(30°)=0.5sin(30°) = 0.5, cos(30°)=ext√3/2cos(30°) = ext{√}3/2, tan(30°)=1/ext√3tan(30°) = 1/ ext{√}3.

  • Application of Concepts

    • Use triangles to find unknown angles and lengths. Utilize altitudes in non-right triangles.
    • Strategies involve dividing the triangle into right triangles when necessary.

  • Law of Cosines

    • Explores generalization of the Pythagorean theorem to non-right triangles.
    • Relationship: a2+b2−2abextcos(C)=c2a^2 + b^2 - 2ab ext{cos}(C) = c^2 and variations for other angles.
    • Areas of squares relate through the areas of rectangles formed by drawn altitudes.

  • Key Takeaway

    • Special right triangles provide a method to find side lengths without needing trigonometric functions or calculators, emphasizing geometric relationships instead.