# 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

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

• 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

• 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.

• Finding the Missing Side Length (press play)