Calculate Semicircle Area Using Thales' Theorem and Similar Triangles

Problem Overview

• 💡 The video aims to calculate the area of a green shaded semicircle within a given geometric diagram.
• 🎯 The setup includes a semicircle with center O, a perpendicular OD, and a chord BC intersecting at point P.
• 📌 Given lengths are CP = 2 units and BP = 3 units, and the primary goal is to find the semicircle's area.

Determining the Radius

• 🔑 To find the area of the semicircle, the radius (r) must first be determined.
• 📐 A line segment AC is connected, forming triangle ABC, where AB is the diameter of the semicircle.
• ✅ By Thales' Theorem, angle ACB is a 90-degree angle, establishing triangle ABC as a right triangle.

Utilizing Similar Triangles

• 🧠 The solution focuses on two right triangles: triangle OBP and triangle ABC.
• 🔬 By assigning complementary angles (alpha and beta), it's demonstrated that both triangles share the same angle measures.
• 🤝 These two triangles are proven to be similar based on the Angle-Angle similarity theorem.

Setting Up Proportions

• 📊 Due to the similarity of triangles OBP and ABC, a proportion is established: AB/BC = PB/OB.
• 📏 The diameter AB is expressed as 2r, and the chord BC is calculated as 3 + 2 = 5 units.
• 🧩 Substituting the known values, the proportion becomes (2r)/5 = 3/r.

Calculating the Area

• 📈 Cross-multiplication of the proportion leads to the equation 2r² = 15, from which r² is found to be 15/2.
• formulae The area of a semicircle is given by (1/2)πr².
• ✨ Plugging in the value of r², the final area of the green shaded semicircle is calculated as 15π/4 square units.