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3.4 – Special Right Triangles

Objectives

  • Discover the ratios of the sides of 45-45-90 triangles and 30-60-90 triangles.
  • Calculate the unknown side length when given one or two side lengths of a 45-45-90 triangle or a 30-60-90 triangle.

 

Key Terms

  • 30-60-90 Triangle – A right triangle with interior angle measures of 30°, 60°, and 90°.
    • In a 30-60-90 triangle, the hypotenuse is always twice as long as the shorter leg and the longer leg is √3 times as long as the shorter leg.
  • 45-45-90 Triangle – An isosceles right triangle with interior angle measures of 45°, 45°, and 90°.
    • In a 45-45-90 triangle, the two legs have the same length and the hypotenuse is √2 times as long as either leg.

 

Notes

45-45-90 Right Triangles
  • 45-45-90 Triangles
    • Formula: Side-Side-Side(√2)
    • Base Unit Formula: 1•1•√2

GeoA 03.04 - 45-45-90 Theorem

GeoA 03.04 - 45-45-90 Chart


  • 45-45-90 Triangles…
    • Are squares, cut in half, diagonally.

GeoA 03.04 - 454590


  • For 45-45-90 Triangles
    • If you know the leg, multiply it by \sqrt{2} to find the hypotenuse.
    • If you know the hypotenuse, divide it by \sqrt{2} to find the legs.

  • Finding the Lengths of the Missing Side Lengths (press play)

 

30-60-90 Right Triangles
  • 30-60-90 Triangles
    • Formula: Side-Side-Side(√3)
    • Base Unit Formula: 1•2•√3

GeoA 03.04 - 30-60-90 Theorem


  • 30-60-90 Triangles
    • The hypotenuse is the longest side of a triangle, but it is NOT the longest leg.  Legs are perpendicular to one another.
    • The longest side (hypotenuse) is always twice as long as the shortest side (short leg).
    • Hypotenuse = 2(Short Leg)
    • Long Leg = √3(Short Leg)

 

  • Parts of a 30-60-90 Triangle
    • The “shorter leg” is the side opposite the 30° angle.
    • The “longer leg” is the side opposite the 60° angle.
    • The hypotenuse is always opposite from the 90° angle.

GeoA 03.04 - 30-60-90 Parts

GeoA 03.04 - EquilateralGeoA 03.04 - Equilateral Bisected


  • Finding the Missing Side Length (press play)

 

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