Notes on the Geometry Problem Presentation

Problem Overview

• Shapes Involved: Semicircle and Quarter Circle
• Key Element: Red tangent line with a length of 5 cm
• Objective: Find the area of a rectangle formed by these shapes.

Step 1: Assigning Variables

• Semicircle Radius: Let the radius be denoted as R.
• Quarter Circle Radius: Denote the radius as X.
• Clarification: The radius of the semicircle (R) is not necessarily equal to the radius of the quarter circle (X).

Step 2: Rectangle Dimensions

• Base of Rectangle:
  • Base = X + 2R
• Height of Rectangle:
  • Height = X
• Area of Rectangle:
  • Formula: Area = Base * Height
  • Area = (X + 2R) * X
  • Area = X² + 2XR

Step 3: Applying Pythagorean Theorem

• Setting Up Right Triangle:
  • Draw a segment from the point of tangency to the center of the semicircle.
  • This segment represents the radius R of the semicircle.
• Right Triangle Configuration:
  • Right angle where the radius intersects the tangent line.
  • Use Pythagorean Theorem:
  • Equation: R² + 5² = (X + R)²
  • Expand and simplify:
    • Left Side: R² + 25
    • Right Side: X² + 2XR + R²
• Canceling R²:
  • This leads to: X² + 2XR = 25

Conclusion: Area of Rectangle

• Since X² + 2XR is equivalent to the area of the rectangle:
  • Area = 25 cm²

Final Note

• Ensure to include the unit in the final answer: cm².