Question

In an equilateral triangle
prove that three times the
square of one side is equal
to four times the square
of one of it's altitudes.​

Answer 1

Answer:

Let ABC be equilateral triangle

let AB= BC= CA = a

As it is equilateral triangle

So, Its perpendicular bisector on any side pass through vertex opposite to it

So, Let AD be Perpendicular bisector of BC passing through A

Now let's find Length of AD

As AD is perpendicular bisector of BC

So, BD = a/2

Now, AD = √(a^2 - (a/2)^2)

= √ ( a^2 - a^2/4)

= √ 3a^2/4

=> √3 a/2

So, According to question

4 × ( AD)^2 => 4 × (√3a/2)^2

= 4 × 3 a^2/4

= 3 a^2

Which is 3 times of square of any side As asked in question

Hence, Proved

#answerwithquality #BAL

Answer 2

Draw an equilateral triangle say, ABC.

Let's the length of each equal side be 'x'

Draw a perpendicular bisector through BC such that, it bisects angle BAC.

Apply the area theorem of equilateral triangle to get area of the triangle.

Now apply Pythagoras theorem to get the altitude.

Explanation:

Refer to the attachment

#answerwithquality

#BAL