Solving the Square Diagonal and Area Problem

This problem involves understanding the relationship between the diagonal and the area of a square. We are given the diagonal of the first square and need to find the diagonal of a second square whose area is three times that of the first.

Understanding the Formulas for a Square

Let \(s\) be the side length of a square and \(d\) be its diagonal. The key formulas for a square are:

• Area \(A = s^2\)
• Diagonal \(d = s\sqrt{2}\)

From these, we can also find the area in terms of the diagonal:

• Since \(s = \frac{d}{\sqrt{2}}\), the Area \(A = s^2 = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{(\sqrt{2})^2} = \frac{d^2}{2}\).

This last formula, \(A = \frac{d^2}{2}\), is very useful when working directly with the diagonal.

Calculating the Area of the First Square

We are given that the diagonal of the first square, \(d_1\), is \(8\sqrt{2}\) cm.

Using the formula \(A_1 = \frac{d_1^2}{2}\):

\(A_1 = \frac{(8\sqrt{2})^2}{2}\)

\(A_1 = \frac{8^2 \times (\sqrt{2})^2}{2}\)

\(A_1 = \frac{64 \times 2}{2}\)

\(A_1 = \frac{128}{2}\)

\(A_1 = 64\) cm².

The area of the first square is 64 cm².

Calculating the Area of the Second Square

The problem states that the area of the second square, \(A_2\), is triple that of the first square.

\(A_2 = 3 \times A_1\)

\(A_2 = 3 \times 64\) cm²

\(A_2 = 192\) cm².

The area of the second square is 192 cm².

Finding the Diagonal of the Second Square

Now we need to find the diagonal, \(d_2\), of the second square, given its area \(A_2 = 192\) cm².

We use the same formula, \(A_2 = \frac{d_2^2}{2}\), and solve for \(d_2\).

\(192 = \frac{d_2^2}{2}\)

Multiply both sides by 2:

\(d_2^2 = 192 \times 2\)

\(d_2^2 = 384\)

Take the square root of both sides to find \(d_2\):

\(d_2 = \sqrt{384}\)

Simplifying the Square Root

To match the options, we need to simplify \(\sqrt{384}\). We look for perfect square factors of 384.

Let's list factors or divide by small prime numbers:

• \(384 \div 2 = 192\)
• \(192 \div 2 = 96\)
• \(96 \div 2 = 48\)
• \(48 \div 2 = 24\)
• \(24 \div 2 = 12\)
• \(12 \div 2 = 6\)

So, \(384 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 6 = 2^6 \times 6\).

Alternatively, we can look for larger perfect squares:

• \(384 = 4 \times 96\)
• \(384 = 16 \times 24\)
• \(384 = 64 \times 6\) (Since \(64 = 8^2\) is a perfect square)

Using \(384 = 64 \times 6\):

\(d_2 = \sqrt{64 \times 6}\)

\(d_2 = \sqrt{64} \times \sqrt{6}\)

\(d_2 = 8\sqrt{6}\) cm.

The diagonal of the second square is \(8\sqrt{6}\) cm.

Checking the Options

The calculated diagonal is \(8\sqrt{6}\) cm, which matches one of the provided options.

Revision Table: Square Formulas

Here's a quick summary of important formulas related to squares:

Additional Information on Square Geometry

Squares are fundamental geometric shapes with many unique properties. They are a special type of rectangle where all sides are equal, and a special type of rhombus where all angles are 90 degrees. The diagonals of a square are equal in length, bisect each other at a 90-degree angle, and also bisect the angles of the square (dividing the 90-degree angles into two 45-degree angles).

Understanding how the diagonal relates to the side using the Pythagorean theorem (\(s^2 + s^2 = d^2 \implies 2s^2 = d^2 \implies d = s\sqrt{2}\)) is key to deriving the area formula in terms of the diagonal (\(A = \frac{d^2}{2}\)). This problem directly applies these relationships.