Right triangles hide secret proportions that unlock geometric puzzles. The altitude to the hypotenuse is the key, acting as a geometric mean between segments. This relationship creates similar triangles, allowing us to solve for unknown lengths.

These proportions aren't just math tricks – they're practical tools. From figuring out building heights to navigating ships, right triangle proportions help us measure and understand the world around us. They're the unsung heroes of geometry in action.

Proportions in Right Triangles

Geometric mean in right triangles

• Square root of the product of two quantities
• In a right triangle, length of the altitude drawn from the right angle to the hypotenuse
• Altitude divides the hypotenuse into two segments
• Length of the altitude is the geometric mean of the lengths of the two hypotenuse segments
• Example: if the hypotenuse segments are 3 and 12, the altitude length is $\sqrt{3 \times 12} = 6$

Altitude and hypotenuse relationships

• Altitude to the hypotenuse creates three similar right triangles
  • Original triangle
  • Triangle formed by the altitude and the shorter hypotenuse segment
  • Triangle formed by the altitude and the longer hypotenuse segment
• Altitude is the mean proportional between the hypotenuse segments
  • $\frac{shorter:segment}{altitude} = \frac{altitude}{longer:segment}$
  • Example: if shorter segment is 4 and altitude is 8, then $\frac{4}{8} = \frac{8}{longer:segment}$, so longer segment is 16
• Altitude squared equals the product of the two hypotenuse segments
  • $altitude^2 = shorter:segment \times longer:segment$
  • Example: if segments are 5 and 20, then $altitude^2 = 5 \times 20 = 100$, so altitude is 10

Proportions in right triangles

• Use proportional relationships among altitude, geometric mean, and hypotenuse segments to solve for unknown lengths
  • Set up proportions using the similar triangles created by the altitude
  • Apply the geometric mean formula to find the length of the altitude or a hypotenuse segment
• Utilize the Pythagorean theorem to find the length of the hypotenuse or a leg
  • $a^2 + b^2 = c^2$, where $c$ is the hypotenuse length and $a$ and $b$ are the leg lengths
  • Example: if legs are 6 and 8, then $6^2 + 8^2 = c^2$, so $36 + 64 = 100$, and $c = 10$

Real-world applications of proportions

• Recognize situations where right triangles and their proportions can be applied
• Construction and architecture
  • Determine the height of a building or structure using its shadow length and angle of elevation
  • Example: a 20 ft pole casts a 15 ft shadow, use tangent to find the sun's angle of elevation
• Navigation and surveying
  • Calculate distances and heights using right triangle properties and proportions
  • Example: measure angle to the top of a cliff and the distance to its base to find the cliff height
• Physics and engineering
  • Analyze forces and vectors in right triangles to solve problems related to motion, equilibrium, and structures
  • Example: find the components of a force vector acting at an angle to a surface