Last updated at May 29, 2018 by

Transcript

Ex 6.5, 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

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.