Technique explanation
This lesson covers two fundamental geometric identities. Pythagoras' Theorem (a² + b² = c²) is derived using altitude-to-hypotenuse similarity. Apollonius' Theorem relates the sides of any triangle to its median: AB² + AC² = 2(AD² + BD²), where AD is the median to side BC.
Pythagoras applies specifically to right triangles, while Apollonius applies to any triangle. Standard fallback: Law of Cosines or coordinate geometry calculations.
Worked examples
Example 1: Apollonius' Theorem
In triangle ABC, AB=5, AC=7, and BC=8. Find median AD.
1. BD = BC/2 = 4.
2. Apply formula: 5² + 7² = 2(AD² + 4²).
3. 25 + 49 = 2(AD² + 16) → 74 = 2(AD² + 16).
4. 37 = AD² + 16 → AD² = 21.
5. Result: AD = √21.
Check: Using Law of Cosines, 5² + 4² - 2(5)(4)cos(B) = 25 + 16 - 20 = 21, which is AD². Verified.
Example 2: Finding a Side
In triangle ABC, AB=3, AC=5, and median AD=4. Find BC.
1. 3² + 5² = 2(4² + BD²).
2. 9 + 25 = 2(16 + BD²) → 34 = 2(16 + BD²).
3. 17 = 16 + BD² → BD² = 1 → BD = 1.
4. Result: BC = 2 × BD = 2.
Check: 3² + 5² = 9 + 25 = 34; formula 2(4² + 1²) = 2(16 + 1) = 34. Verified.
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