Problem 78 - Algebra A Combined Function

Question

Solve.The top of a square coffee table has a diagonal that measures 30 inches. Find the length of each side of the top of the coffee table.

Step-by-Step Solution

Answer

Each side length is \( 15\sqrt{2} \) inches.

Step 1: Understand the Problem

We have a square coffee table, and we know the diagonal length is 30 inches. We need to find the length of each side of the square.

Step 2: Recall the Pythagorean Theorem for Squares

The diagonal of a square divides it into two congruent right-angled triangles. If each side of the square is denoted by \( s \), then the relationship between the sides and the diagonal \( d \) is given by the Pythagorean theorem: \( d^2 = s^2 + s^2 \).

Step 3: Set Up the Equation

Since the diagonal \( d = 30 \) inches, we substitute into the formula: \( 30^2 = s^2 + s^2 \), which simplifies to \( 900 = 2s^2 \).

Step 4: Solve for \( s^2 \)

To isolate \( s^2 \), divide both sides by 2: \( s^2 = \frac{900}{2} = 450 \).

Step 5: Solve for \( s \)

To find \( s \), take the square root of 450: \( s = \sqrt{450} \). Simplifying, \( \sqrt{450} = \sqrt{25 \times 18} = 5 \times \sqrt{18} \). Further simplifying, \( \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \). Hence, \( s = 15\sqrt{2} \).

Step 6: Conclusion

The side length of the square coffee table is \( 15\sqrt{2} \) inches.

Key Concepts

Square Geometry

A square is a unique geometric shape where all four sides are of equal length. This makes it a special type of rectangle with additional properties. In square geometry:

• All angles are right angles, meaning they are each 90 degrees.
• The opposite sides are parallel.
• The diagonals are equal in length and bisect each other at right angles, dividing the square into two equal right-angled triangles.

Understanding these attributes helps solve problems related to squares, such as calculating the length of sides when the diagonal is known. In everyday contexts, squares can be seen in various applications, from design patterns to architecture.

Diagonal of a Square

The diagonal of a square is a crucial geometric feature that can be quite useful in calculations. When you draw a diagonal in a square, it splits the square into two equal right-angled triangles.
To calculate the length of a side of a square when you know the diagonal, you can use the relationship derived from the Pythagorean theorem:

• Let the side of the square be denoted by \( s \).
• If the diagonal is denoted by \( d \), according to the Pythagorean theorem, \( d^2 = s^2 + s^2 \).

This formula arises because the diagonal acts as the hypotenuse of the right-angled triangles formed. Therefore, if you know the diagonal, you can easily calculate the side of the square by rearranging the formula to solve for \( s \). This principle is highly applicable in fields like robotics and computer graphics where calculating distances in square layouts is essential.

Right-Angled Triangles

Right-angled triangles are foundational in geometry due to their easily understood properties, which are perfectly exemplified in squares. The right angle in a triangle means there is one angle of 90 degrees. This right angle leads to the establishment of many useful theorems, the most famous being the Pythagorean theorem. In the context of a square:

• The diagonal divides the square into two right-angled triangles.
• Each leg of the right triangle corresponds to a side of the square.
• The hypotenuse of such a triangle is the diagonal itself.

The Pythagorean theorem states that for a right-angled triangle with legs of length \( a \) and \( b \), and hypotenuse \( c \), the relation \( a^2 + b^2 = c^2 \) holds. In a square, both legs are equal, thus simplifying the calculations to find unknown lengths. This theorem not only solves geometric puzzles but is also used in practical applications like construction and navigation.