It seems like the question is asking for the ratio between the side length $a$ of an equilateral triangle and its height $h$. Let's solve this step by step.

Step 1: Formula for the height of an equilateral triangle

For an equilateral triangle with side length $a$, the height $h$ can be given by:

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

Step 2: Find the ratio $\frac{a}{h}$

We need to find the ratio of the side length $a$ to the height $h$:

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

Step 3: Simplify the ratio

To simplify $\frac{2}{\sqrt{3}}$, we multiply the numerator and denominator by $\sqrt{3}$ to rationalize the denominator:

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

Step 4: Interpret the result in terms of the options given

The simplified ratio $\frac{2\sqrt{3}}{3}$ does not directly match the options provided. However, the options might be written differently, possibly due to a misprint or error. None of the provided options seem to directly correspond to the correct ratio, but if there's a minor printing error, (D) $2:43$ might be a misrepresentation of $\frac{2\sqrt{3}}{3}$.

Conclusion:

Given the correct formula and simplification, the expected ratio of the side length to height should be $\frac{2\sqrt{3}}{3}$. Since none of the options match this exactly, this might indicate an issue with the options given in the question.

Do you have any questions or need further clarification?

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Here are five related questions:

1. What is the area of an equilateral triangle with side length $a$?
2. How does the height of an equilateral triangle relate to its circumradius?
3. How do you derive the formula for the height of an equilateral triangle?
4. What is the relationship between the side length and the inradius of an equilateral triangle?
5. How does the formula for the height change if the triangle is not equilateral?

Tip: Always double-check mathematical options in multiple-choice questions for possible misprints or errors in presentation.