If the area of a triangular sign board shaped like an equilateral triangle is $144\sqrt{3}$ cm², what are its side length and height?

The correct option is Side = 24 cm, Altitude = 12$\sqrt{3}$ cm.

Step-by-step explanation:

Step 1: Formula for the area of an equilateral triangle
The area of an equilateral triangle with side length a is calculated using the standard formula:

$\text{Area}=\frac{\sqrt{3}}{4}{a}^2$

Step 2: Calculate the side length (a)
We are given that the area of the triangular sign board is $144\sqrt{3}$ cm². Substituting this value into the area formula gives:

$\frac{\sqrt{3}}{4}{a}^2=144\sqrt{3}$

Divide both sides of the equation by $\sqrt{3}$:

$\frac{1}{4}{a}^2=144$

Multiply both sides by 4:

$a^2=144\times 4$

$a^2=576$

Taking the square root on both sides:

$a=\sqrt{576}=24\text{cm}$

So, the side length of the equilateral triangle is 24 cm.

Step 3: Calculate the height/altitude (h)
The altitude h of an equilateral triangle with side length a is given by the formula:

$h=\frac{\sqrt{3}}{2}a$

Substituting $a=24$ cm into the formula:

$h=\frac{\sqrt{3}}{2}\times 24$

$h=12\sqrt{3}\text{cm}$

Conclusion:
The side length of the triangle is 24 cm and its altitude is 12$\sqrt{3}$ cm.