A square and an equilateral triangle are drawn on the same base. The ratio of their area is

Options

A. `2:1`
B. `1:1`
C. `sqrt(3):4`
D. `4 : sqrt(3)`

Answer

To find the ratio of the areas of a square and an equilateral triangle drawn on the same base, we can follow these steps:

### Step 1: Define the side length
Let the side length of the square and the base of the equilateral triangle be denoted as \( a \).

### Step 2: Calculate the area of the square
The area \( A_{\text{square}} \) of the square is given by the formula:
\[ A_{\text{square}} = a^2 \]

### Step 3: Calculate the height of the equilateral triangle
For an equilateral triangle with side length \( a \), the height \( h \) can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]

### Step 4: Calculate the area of the equilateral triangle
The area \( A_{\text{triangle}} \) of the equilateral triangle is given by the formula:
\[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times h \]
Substituting the height from Step 3:
\[ A_{\text{triangle}} = \frac{1}{2} \times a \times \left(\frac{\sqrt{3}}{2} a\right) = \frac{\sqrt{3}}{4} a^2 \]

### Step 5: Find the ratio of the areas
Now, we can find the ratio of the area of the square to the area of the equilateral triangle:
\[ \text{Ratio} = \frac{A_{\text{square}}}{A_{\text{triangle}}} = \frac{a^2}{\frac{\sqrt{3}}{4} a^2} \]
The \( a^2 \) terms cancel out:
\[ \text{Ratio} = \frac{1}{\frac{\sqrt{3}}{4}} = \frac{4}{\sqrt{3}} \]

### Step 6: Simplify the ratio
Thus, the ratio of the area of the square to the area of the equilateral triangle is:
\[ \text{Ratio} = 4 : \sqrt{3} \]

### Final Answer
The ratio of the areas of the square and the equilateral triangle is \( 4 : \sqrt{3} \).

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