Question

A traffic signal board indicating 'SCHOOL AHEAD' is an equilateral triangle with side 2a, then area of the traffic signal is:
a) √4/2a
b) √3/4a²
c) √4/2a²
d) √30a²​

Answer 1

*replace b and use a (related to attachment)

Side = 2a units

We know, area of ∆ = 1/2 bh

And, perpendicular from opposite vertex of an equilateral triangle bisects the opposite side. (You can prove it through RHS congruence criteria.)

By Pythagoras theorem,

h² + b² = (2a)²

=> h² = 4a² - a² = 3a²

=> h² = √(3a²)

=> h = ± a√3 units = + a√3 units

So, area = 1/2 × b × h = 1/2 × 2a × a√3 = a²√3 or √3·a² (d).

Aliter:

• Area of ∆ = 1/2 ab sinC = 1/2 (2a)²(sin60°) = 2a² × (√3)/2 = a²√3
• you can use the heron's formula to obtain the answer too.

Area = s(s - a)(s - b)(s - c)

where, a, b, c are sides

& s = semi perimeter = (a + b + c)/2

Therefore, s = 3a

So, area = √[3a(3a - 2a)³] = √[3a(a)³] = √[3a⁴] = a²√3.

• For sake of shortcut: directly put side value in (√3)/4 × a² where a is side.