Find the height of an equilateral triangle having side 2a..

Explanation:- 

The given problem states that ABC is an equilateral triangle, and we are required to find the height of the triangle. Since ABC is an equilateral triangle, all sides are equal in length.

Let us draw a perpendicular line AD from vertex A to side BC. Since ABC is an equilateral triangle, AD is the perpendicular bisector of BC. This means that BD = DC = ½ BC = ½a, where a is the length of each side of the equilateral triangle.

Now, let us apply the Pythagorean theorem in ∆ABD. According to the theorem, the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side (hypotenuse). In this case, AB is the hypotenuse, and AD and BD are the two shorter sides.

Therefore, we have:

AB^2 = AD^2 + BD^2

Substituting the values of AB and BD, we get:

(2a)^2 = AD^2 + (1/2a)^2

Simplifying, we get:

4a^2 = AD^2 + 1/4a^2

Multiplying both sides by 4a^2, we get:

16a^4 = 4a^2AD^2 + 1

Rearranging, we get:

4a^2AD^2 = 16a^4 – 1

Taking the square root of both sides, we get:

2aAD = √(16a^4 – 1)

Dividing by 2a, we get:

AD = √(16a^2 – 1)/2

Since the area of an equilateral triangle is (sqrt(3)/4)a^2, and the area is also given by (1/2)bh, where b is the base (a in this case) and h is the height, we can equate these two expressions and solve for h:

(sqrt(3)/4)a^2 = (1/2)ah

Simplifying, we get:

h = (2/√3)a

Substituting the value of AD in terms of a, we get:

h = AD = √(16a^2 – 1)/2

Therefore, we have:

√(16a^2 – 1)/2 = (2/√3)a

Squaring both sides and simplifying, we get:

a^2 = 3/4

Therefore, a = √(3/4) = √3/2

Substituting the value of a in the expression for h, we get:

h = (2/√3)a = 2/√3 × √3/2 = √3

Hence, the height of the equilateral triangle is √3 units.