Question

in a equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Answer 1

in equilateral triangle altitude=√3a/2. If one of sides is a.
You can verify altitude measurement by Pythagoras theorem.
Just substitute the values according to the question.
You will get answer as below in the pic.

Answer 2

$\bf\huge\underline{Question}$⠀

In a equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

$\bf\huge\underline{Solution}$

We have an equilateral ∆ABC, in which AD _|_ BC. Since, an altitude is an equilateral ∆, bisects the corresponding side.

Therefore, D is the mid point of BC

=> BD = DC ⠀⠀⠀⠀⠀⠀⠀⠀⠀[each$\dfrac{1}{2}$BC]

In right ∆ADB,

⠀⠀AB² = AD² + BD²

⠀⠀⠀⠀⠀⠀⠀⠀[using Pythagoras theorem]

=> AD² + ($\dfrac{1}{2}$BC)² = AD² + $\dfrac{1}{4}$BC²

=> 4AB² = 4AD² + BC²

=> 4AB² = 4 AD² + AB² ⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀[AB = BC = AC]

=> 4AD² = 4AB² - AB² or 3AB² = 4AD²