The ratio of the ares of two equilateral triangles is `16 : 9` . If the perimeter of the smaller traingle is 63 cm, then how much larger is a side of the larger triangle is a side of the larger triangel than a side of the smaller triangle ?

Options

A. 3 cm
B. 7 cm
C. 4 cm
D. 5 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Relationship Between Areas and Sides
The area of an equilateral triangle is given by the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
where \( s \) is the length of a side of the triangle.

### Step 2: Set Up the Ratio of Areas
Given the ratio of the areas of two equilateral triangles is \( 16:9 \), we can express this as:
\[ \frac{\text{Area of larger triangle}}{\text{Area of smaller triangle}} = \frac{16}{9} \]
Let \( A \) be the side of the larger triangle and \( B \) be the side of the smaller triangle. Thus:
\[ \frac{\frac{\sqrt{3}}{4} A^2}{\frac{\sqrt{3}}{4} B^2} = \frac{16}{9} \]
This simplifies to:
\[ \frac{A^2}{B^2} = \frac{16}{9} \]

### Step 3: Take the Square Root
Taking the square root of both sides gives:
\[ \frac{A}{B} = \frac{4}{3} \]

### Step 4: Express the Side of the Smaller Triangle
Since the perimeter of the smaller triangle is given as 63 cm, and the perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3s \]
For the smaller triangle:
\[ 3B = 63 \implies B = \frac{63}{3} = 21 \text{ cm} \]

### Step 5: Find the Side of the Larger Triangle
Using the ratio we found earlier:
\[ A = \frac{4}{3} B = \frac{4}{3} \times 21 = 28 \text{ cm} \]

### Step 6: Calculate the Difference in Side Lengths
Now, we need to find how much larger the side of the larger triangle is than the side of the smaller triangle:
\[ A - B = 28 - 21 = 7 \text{ cm} \]

### Final Answer
The side of the larger triangle is 7 cm larger than the side of the smaller triangle.

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