Solving an Olympiad Math Problem: Area of a Right Triangle

Problem Overview

• 💡 The video aims to calculate the area of a green shaded right triangle ABC.
• 📌 Given information includes one side length, AB = 13 units, and the condition that all side lengths are positive integers.
• 🎯 The primary goal is to find the area of this specific triangle.

Utilizing Pythagorean Theorem

• 📐 The area of a triangle formula is half times base times height.
• 🔑 The Pythagorean theorem (a² + b² = c²) is applied, where side AB is 13, and the other two sides are labeled B (base) and C (hypotenuse).
• 📝 This leads to the equation 13² + b² = c², which simplifies to c² - b² = 169.

Solving for Unknown Side Lengths

• 🧩 The difference of two squares identity (a² - b² = (a + b)(a - b)) is used to factor c² - b² into (c + b)(c - b) = 169.
• 🔢 By considering 169 as a product of two factors (169 * 1), a system of two linear equations is formed: c + b = 169 and c - b = 1.
• ✅ Solving these equations simultaneously yields c = 85 and b = 84, confirming they are positive integers.

Calculating the Triangle's Area

• 📊 With the base b = 84 units and the height AB = 13 units determined, the area formula is applied.
• ✨ The area is calculated as (1/2) * 84 * 13.
• 🏆 The final area of the green shaded triangle is found to be 546 square units.