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3.4 – Special Right Triangles
Objectives
 Discover the ratios of the sides of 454590 triangles and 306090 triangles.
 Calculate the unknown side length when given one or two side lengths of a 454590 triangle or a 306090 triangle.
Key Terms
 306090 Triangle – A right triangle with interior angle measures of 30°, 60°, and 90°.
 In a 306090 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.
 454590 Triangle – An isosceles right triangle with interior angle measures of 45°, 45°, and 90°.
 In a 454590 triangle, the two legs have the same length and the hypotenuse is √2 times as long as either leg.
Notes
454590 Right Triangles 
 454590 Triangles
 Formula: SideSideSide(√2)
 Base Unit Formula: 1•1•√2
 454590 Triangles…
 Are squares, cut in half, diagonally.
 For 454590 Triangles
 If you know the leg, multiply it by to find the hypotenuse.
 If you know the hypotenuse, divide it by to find the legs.
 Finding the Lengths of the Missing Side Lengths (press play)

306090 Right Triangles 
 306090 Triangles
 Formula: SideSideSide(√3)
 Base Unit Formula: 1•2•√3
 306090 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 306090 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.
 Finding the Missing Side Length (press play)

Important!
Practice (Apex Study 3.4)
 Practice: Pgs 8, 16, 19, 21
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