Understanding Square Area and Perimeter Ratios

This problem asks us to find the ratio of the perimeters of two squares, given the ratio of their areas. We need to understand how the area and perimeter of a square relate to its side length.

For any square:

• Area (A) = side length (s) \(\times\) side length (s) = \(s^2\)
• Perimeter (P) = 4 \(\times\) side length (s) = \(4s\)

Step-by-Step Solution to Find Perimeter Ratio

Let the two squares be Square 1 and Square 2.

Let the side length of Square 1 be \(s_1\) and its area be \(A_1\).

Let the side length of Square 2 be \(s_2\) and its area be \(A_2\).

We are given the ratio of their areas: \(A_1 : A_2 = 16 : 9\).

This can be written as:

\(\frac{A_1}{A_2} = \frac{16}{9}\)

Substitute the area formula (\(A = s^2\)) for each square:

\(\frac{s_1^2}{s_2^2} = \frac{16}{9}\)

To find the ratio of the side lengths (\(s_1 : s_2\)), we need to take the square root of both sides of the equation:

\(\sqrt{\frac{s_1^2}{s_2^2}} = \sqrt{\frac{16}{9}}\)

\(\frac{s_1}{s_2} = \frac{\sqrt{16}}{\sqrt{9}}\)

\(\frac{s_1}{s_2} = \frac{4}{3}\)

So, the ratio of the side lengths of the two squares is \(s_1 : s_2 = 4 : 3\).

Now, let's find the ratio of their perimeters. The perimeter of Square 1 is \(P_1 = 4s_1\) and the perimeter of Square 2 is \(P_2 = 4s_2\).

The ratio of their perimeters is \(P_1 : P_2 = 4s_1 : 4s_2\).

We can simplify this ratio by dividing both parts by 4:

\(P_1 : P_2 = s_1 : s_2\)

Since we found that \(s_1 : s_2 = 4 : 3\), the ratio of the perimeters is \(4 : 3\).

Now let's check the given options. The ratio \(4 : 3\) is equivalent to \(16 : 12\), because \(\frac{16}{12} = \frac{4 \times 4}{3 \times 4} = \frac{4}{3}\).

Therefore, the ratio of their perimeters is \(16 : 12\).

Analyzing the Options

Let's look at the provided options based on our calculation:

• Option 1: 9 : 16 (This is the inverse of the area ratio)
• Option 2: 9 : 12 (Incorrect)
• Option 3: 12 : 16 (Incorrect, this is the inverse ratio of 16:12)
• Option 4: 16 : 12 (This ratio is equivalent to 4:3, which is the ratio of side lengths and perimeters)

Our calculated ratio \(4:3\) is equivalent to \(16:12\), which matches Option 4.

Revision Table: Square Properties

Additional Information: Ratios and Proportions

A ratio is a comparison of two quantities. If the ratio of quantity A to quantity B is \(a : b\), it can also be written as a fraction \(\frac{a}{b}\).

When dealing with areas and lengths of similar shapes (all squares are similar), the ratio of their areas is the square of the ratio of their corresponding side lengths. Conversely, the ratio of their side lengths is the square root of the ratio of their areas.

The ratio of perimeters of similar shapes is the same as the ratio of their corresponding side lengths because perimeter is a linear measurement.

In this problem:

• Ratio of Areas = (\(Ratio\; of\; Sides\))^2
• Ratio of Sides = \(\sqrt{Ratio\; of\; Areas}\)
• Ratio of Perimeters = Ratio of Sides

Given Area Ratio \(16 : 9\), Side Ratio is \(\sqrt{16} : \sqrt{9} = 4 : 3\).

Perimeter Ratio is the same as Side Ratio, which is \(4 : 3\). The option \(16 : 12\) is equivalent to \(4 : 3\).