Problem 50 - Algebra 2 and Trigonometry

Question

What is the length of a side of a square if the length of a diagonal is \(\sqrt{72}\) inches?

Step-by-Step Solution

Answer

The length of a side of the square is 6 inches.

Step 1: Understand the Relationship Between Side and Diagonal of a Square

The diagonal of a square creates two right-angled triangles within the square. According to the Pythagorean theorem, the relationship in a square between each side (s) and the diagonal (d) is given by: \( d = s\sqrt{2} \).

Step 2: Substitute the Given Diagonal Length

We know that the diagonal \( d = \sqrt{72} \). Substituting the diagonal in the equation \( d = s\sqrt{2} \) gives us \( \sqrt{72} = s\sqrt{2} \).

Step 3: Isolate the Side Length Equation

To find \( s \), divide both sides of the equation by \( \sqrt{2} \): \( s = \frac{\sqrt{72}}{\sqrt{2}} \).

Step 4: Simplify the Fraction

Simplify \( \frac{\sqrt{72}}{\sqrt{2}} \) using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), which gives \( s = \sqrt{\frac{72}{2}} = \sqrt{36} \).

Step 5: Calculate the Value of the Side

The square root of 36 is 6, so \( s = 6 \). This means the length of a side of the square is 6 inches.

Key Concepts

Diagonal of a Square

When we talk about the diagonal of a square, it might first seem a bit enigmatic. However, understanding its properties unlocks a clear view of this geometric figure. A diagonal in a square is a line segment that connects opposite corners. When you draw it, you split the square into two identical right-angled triangles.

The magical thing about the diagonal is its relationship with the sides of the square. According to the Pythagorean theorem, the diagonal, denoted as \( d \), is related to the side of the square, denoted as \( s \), by the formula:

• \( d = s\sqrt{2} \)

This formula tells us that the length of the diagonal is always the length of the side multiplied by the square root of 2. This happens because, in the right-angled triangle formed with the diagonal as the hypotenuse, both other sides (legs) are actually sides of the square.

Understanding this relationship helps in solving problems that ask you to find the length of the sides or the diagonal, as you can easily switch between them using this formula.

Right-Angled Triangle

A right-angled triangle is one of the most interesting figures in geometry. It is called 'right-angled' because one of its angles is a right angle, which measures 90 degrees. In the case of a square, drawing a diagonal automatically slices it into two right-angled triangles.

In a right-angled triangle:

• The longest side is called the 'hypotenuse', which, in our case, is the diagonal of the square.
• The other two sides are called 'legs', which, in our case, are the sides of the square.

The Pythagorean theorem applies here, stating that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides. Understanding this potently powerful theorem allows you to unveil distances and measurements that seem tricky at first glance. With a basic knowledge of the theorem, you can explore and solve a myriad of geometric problems.

Simplifying Square Roots

Simplifying square roots can sometimes seem daunting, but with some practice, it becomes much easier. When you have an expression with a square root, like \( \sqrt{72} \), it might not seem straightforward at first to simplify it.To simplify a square root, look for the factors of the number under the root and see if any of them are perfect squares. For example, with \( \sqrt{72} \):

• Break it down: \( 72 = 36 \times 2 \)
• Since 36 is a perfect square, it simplifies to \( \sqrt{36} = 6 \)
• Thus, \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \)

This simplification is key when you are plugging square roots back into equations. By simplifying, you make equations easier to handle and solutions easier to find. The more you practice simplifying, the more intuitive it becomes, enabling you to solve problems swiftly and with confidence.