Question 8 - Pythagoras Theoram - Proving - Chapter 6 Class 10 Triangles

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Question 16 In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes. Given:- Equilateral triangle ABC with each side a & AD as one of its altitudes To Prove :- 3 × Square of one side = 4 × square of one of it’s altitude ⇒ 3a2 = 4AD2 Proof:- In Δ ADB & Δ ADC AB = AC AD = AD ∠ ADB=∠ ADC Hence ∆ ADB ≅ ∆ ADC Hence , BD = DC BD = DC = 1/2BC BD = DC = 𝑎/2 Now, ADB is a right angle triangle Using Pythagoras theorem (Hypotenuse)2 = (Height)2 + (Base)2 (AB)2 = (AD)2 + (BD)2 (a)2 = AD2 + (1/2 𝑎)2 a2 = AD2 + 𝑎2/4 a2 – 𝑎2/4=𝐴𝐷2 (4𝑎2 − 𝑎2)/4=𝐴𝐷2 3𝑎2/4=𝐴𝐷2 3a2 = 4 ×𝐴𝐷2 3a2 = 4AD2 Hence proved