Geometry Lesson 7.4: Similarity and Right Triangles

Essential Question

• What is the relationship between the altitude to the hypotenuse, triangle similarity, and the geometric mean in a right triangle?

Goals

• Use similarity and the geometric mean to solve problems involving right triangles.

Key Vocabulary

• Geometric Mean: A type of mean or average, which indicates the central tendency of a set of numbers using the product of their values.

Key Concepts

Altitude to Hypotenuse Theorem

• Theorem 7.4: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original and to each other.
• When you draw an altitude from the right angle to the hypotenuse, it forms three similar right triangles.
  • The original large triangle and two smaller triangles on either side of the altitude.
• Similarity: Verified using the Angle-Angle Similarity Theorem because the triangles share two congruent angles.

Solving for Missing Sides

• Use the similarity of triangles to set up proportions and solve for missing side lengths.
• Example 1: Solving for sides in a divided triangle using similarity statements and cross-multiplication.
  • X and Y Calculation: Use similarity to form proportions and solve using cross-multiplication.

Geometric Mean in Right Triangles

• Geometric Mean: For triangles, it is the square root of the product of two segments.
• Application:
  • CD (altitude) is the geometric mean of AD and DB (segments of the hypotenuse).
  • Corollary 1 & 2:
    • AC is the geometric mean of AD and the total hypotenuse.
    • BC is the geometric mean of DB and the total hypotenuse.

Problem Examples

• Example 2: Steps to find segment lengths using geometric means.
  • Set up proportions using similarity and solve for missing lengths.
  • Use calculations involving the product of known sides and their square roots to find unknown sides.

Practice Problems

• Solving for X and Y:
  • Use geometric means and proportions to find missing side lengths in right triangles.
  • X Calculation: Use altitude as a geometric mean of hypotenuse segments.
  • Y Calculation: Use leg as a geometric mean to solve for unknowns.
• Finding Z: Utilizes closest part and hypotenuse to solve using geometric mean.

Conclusion

• Understanding the relationship between altitude, similarity, and geometric mean helps solve right triangle problems.
• Utilize theorems and principles from similarity to efficiently calculate unknown measurements.