The ratio of the areas of two equilateral triangles is 16:9. If the perimeter of the smaller triangle is 63 cm, then how much larger is a side of the larger triangle 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 step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the ratio of the areas of the triangles
The ratio of the areas of two equilateral triangles is given as 16:9.

### Step 2: Relate the ratio of the areas to the sides of the triangles
For equilateral triangles, the ratio of the areas is equal to the square of the ratio of their sides. If we denote the side of the larger triangle as \( A \) and the side of the smaller triangle as \( a \), we can write:

\[ \frac{Area_{larger}}{Area_{smaller}} = \frac{A^2}{a^2} = \frac{16}{9} \]

### Step 3: Take the square root of the ratio
Taking the square root of both sides gives us the ratio of the sides:

\[ \frac{A}{a} = \frac{4}{3} \]

### Step 4: Find the side length of the smaller triangle
We know the perimeter of the smaller triangle is 63 cm. The perimeter \( P \) of an equilateral triangle is given by:

\[ P = 3a \]

Setting this equal to 63 cm, we can solve for \( a \):

\[ 3a = 63 \implies a = \frac{63}{3} = 21 \text{ cm} \]

### Step 5: Find the side length of the larger triangle
Using the ratio of the sides we found earlier, we can express \( A \) in terms of \( a \):

\[ \frac{A}{21} = \frac{4}{3} \]

Now, solving for \( A \):

\[ A = 21 \times \frac{4}{3} = 28 \text{ cm} \]

### Step 6: Calculate the difference in side lengths
To find out how much larger the side of the larger triangle is than that of the smaller triangle, we subtract \( a \) from \( A \):

\[ A - a = 28 - 21 = 7 \text{ cm} \]

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

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