Problem 26 - Intermediate Algebra

Question

Find the missing side lengths in each triangle. Give the exact answer and then an approximation to two decimal places when appropriate. See Example 2. One side of a square is \(5 \frac{1}{2}\) in. long. Find the length of its diagonal. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

The diagonal is √60.5; approximately 7.78 inches.

Step 1: Understand the Relationship in a Square

In a square, all sides are equal, and the diagonal forms the hypotenuse of two congruent right triangles within the square. Each side of the square is a leg of these triangles.

Step 2: Set Up the Formula

To find the diagonal of a square, use the Pythagorean theorem. If the side of the square is \(a\), then the diagonal \(d\) can be calculated using: \(d = \sqrt{a^2 + a^2} = \sqrt{2a^2}\).

Step 3: Substitute the Given Side Length

Substitute \(a = 5 \frac{1}{2}\) into the formula. First, convert the mixed number to an improper fraction: \(5 \frac{1}{2} = \frac{11}{2}\). Now substitute to find the diagonal, \(d = \sqrt{2 \left(\frac{11}{2}\right)^2}\).

Step 4: Calculate the Diagonal

Compute the square of the side: \(\left(\frac{11}{2}\right)^2 = \frac{121}{4}\). Then compute the diagonal: \(d = \sqrt{2 \times \frac{121}{4}} = \sqrt{\frac{242}{4}} = \sqrt{60.5}\).

Step 5: Simplify and Approximate the Answer

The exact diagonal is \(\sqrt{60.5}\). Using a calculator, approximate \(\sqrt{60.5} \approx 7.78\). Thus, the diagonal's length is approximately 7.78 inches.

Key Concepts

Understanding Geometry

Geometry is all about understanding the properties and relations of points, lines, surfaces, and solids. It's like the blueprint of shapes around us. For instance, in this exercise, we are working with a square. A square is a simple geometric shape with four equal sides and four right angles. It's the basis for understanding more complex shapes and calculating areas, perimeters, and even diagonal lengths, as we'll see.

When dealing with geometric shapes, the tools we often use include:

• Lines (straight and precise paths between two points)
• Angles (formed when two lines meet)
• Shapes (like triangles, circles, and squares)

Understanding geometry helps us determine dimensions and solve puzzles about measurements and space.

Properties of a Square

A square is a special type of rectangle where all sides are equal in length. Each angle in a square is a right angle, meaning it measures 90 degrees.

Some useful properties of squares are:

• All sides are equal: This means if one side is known, all are the same.
• Diagonals are equal: Each diagonal splits the square into two congruent right triangles.
• The diagonals intersect at 90 degrees and bisect each other exactly.

These properties make squares incredibly symmetrical and predictable, which is why they are ideal for exercises involving diagonal calculations, like using the Pythagorean theorem in this exercise.

Understanding Triangle Properties

When a square is divided by its diagonals, it forms two congruent right triangles. These triangles have unique properties that are fundamental to solving problems like this one.

The right triangle's properties include:

• One angle is always 90 degrees.
• The hypotenuse is the longest side, opposite the right angle.
• The two shorter sides, known as legs, form the right angle.
• The Pythagorean Theorem helps us relate the lengths of these sides: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

By using these properties, we can determine unknown lengths in many geometric problems. In the square's context, notice that the diagonal runs through as the hypotenuse of the right triangle. Calculating it requires knowing the side's length, which acts as each triangle's two legs in the Pythagorean formula.