8.2 Special right triangles

Special right triangles are super useful in geometry. They have fixed angle measures and side length ratios that make calculations easier. 45-45-90 and 30-60-90 triangles pop up all over the place in math and real life.

These triangles are key for solving problems in construction, navigation, and design. Knowing their properties helps you quickly find missing side lengths or angles without complex math. They're like geometric shortcuts you'll use again and again.

Properties and Applications of Special Right Triangles

Properties of 45-45-90 triangles

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• Consist of two 45° angles and one 90° angle
• Two legs (sides opposite 45° angles) are congruent have equal length
• Hypotenuse is $\sqrt{2}$ times length of a leg
  • Leg length $x$, hypotenuse length $x\sqrt{2}$ (if leg is 1, hypotenuse is $\sqrt{2}$)
• Find leg length given hypotenuse: divide hypotenuse by $\sqrt{2}$
  • Hypotenuse length $h$, leg length $\frac{h}{\sqrt{2}}$ (if hypotenuse is $2\sqrt{2}$, leg is 2)
• Diagonal of a square forms two 45-45-90 triangles
• Used in construction, architecture, and engineering (roof pitch, corner braces)

Properties of 30-60-90 triangles

• Contain one 30° angle, one 60° angle, and one 90° angle
• Shortest side opposite 30° angle, medium side opposite 60° angle, longest side (hypotenuse) opposite 90° angle
• Shortest side is half length of hypotenuse
  • Hypotenuse length $h$, shortest side length $\frac{h}{2}$ (if hypotenuse is 2, shortest side is 1)
• Medium side is $\sqrt{3}$ times length of shortest side
  • Shortest side length $x$, medium side length $x\sqrt{3}$ (if shortest side is 1, medium side is $\sqrt{3}$)
• Hypotenuse is twice length of shortest side
  • Shortest side length $x$, hypotenuse length $2x$ (if shortest side is $\sqrt{3}$, hypotenuse is $2\sqrt{3}$)
• Appear in equilateral triangles, regular hexagons
• Applied in navigation, surveying, and computer graphics (30° and 60° angles)

Special triangles in real-world contexts

• Determine special right triangle type from given angle measures or side length ratios
• Apply relevant 45-45-90 or 30-60-90 triangle properties to find missing side lengths
• Verify calculated side lengths using Pythagorean theorem ($a^2 + b^2 = c^2$)
• Interpret results in real-world context
  • Calculated length should be positive value in appropriate units (meters for building height)
• Examples:
  1. Determine height of a flag pole from its shadow length and angle of elevation (30-60-90)
  2. Calculate diagonal length of a square room to install a support beam (45-45-90)
  3. Find the length of a ramp needed to reach a certain height with a 30° angle (30-60-90)