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

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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