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Understanding Special Right Triangles: 45°-45°-90° and 30°-60°-90° Properties

This guide explores the properties and relationships of special right triangles, specifically the 45°-45°-90° and 30°-60°-90° triangles. It discusses how the legs of the 45° triangle are congruent, with the hypotenuse being the leg length multiplied by √2. For the 30°-60°-90° triangle, it describes how the hypotenuse is twice the shorter leg, and the longer leg is the shorter leg multiplied by √3. Key example calculations are provided to demonstrate these concepts using specific leg measurements.

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Understanding Special Right Triangles: 45°-45°-90° and 30°-60°-90° Properties

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  1. Special Right Triangles 45° 45° 90° and 30° 60° 90°

  2. 45° Y X 45° X Z The legs are congruent because sides opposite congruent angles are congruent

  3. Legs are congruent Hypotenuse = leg times the √2 hypotenuse leg leg

  4. Y 12ft 8in X 45° 45° X Y X= 6√2 ft X= 8in Y= 6√2 ft Y= 8√2 in

  5. 2X 2X Y X 2X The two right triangles are congruent so Z=2X 60° Z Z 60° 60° Z

  6. Hypotenuse = 2 short legs Long leg = short leg • 3 Hypotenuse Long Leg 60° Short Leg

  7. Y 16 X Y 60° 60° 7 X 16=2X X = 7√3 Y = 14 X = 8 Y = 8√3

  8. Y Y 5√3 9 60° 60° X X 5√3=X√3 X = 3√3 Y = 6√3 X = 5 Y = 10

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