Problem 26 - Prealgebra

Question

A The floor exercise mat measures 40 feet by 40 feet. Find the measure of the diagonal.

Step-by-Step Solution

Answer

The diagonal of the mat measures approximately 56.57 feet.

Step 1: Understand the Problem

We need to find the diagonal of a square mat with each side measuring 40 feet. This can be solved using the Pythagorean Theorem since the diagonal forms a right triangle with two sides of the square.

Step 2: Recall the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides. The formula is: \[ a^2 + b^2 = c^2 \] where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

Step 3: Set Up the Equation

Since the mat is a square, the sides \( a \) and \( b \) are each 40 feet. Substitute these values into the Pythagorean Theorem: \[ 40^2 + 40^2 = c^2 \]

Step 4: Calculate Each Square

Calculate \( 40^2 \) which is 1600. Therefore, the equation becomes: \[ 1600 + 1600 = c^2 \]

Step 5: Solve for the Hypotenuse

Add the two squares: \[ 1600 + 1600 = 3200 \] Now, solve for \( c \) by taking the square root of 3200: \[ c = \sqrt{3200} \]

Step 6: Simplify the Square Root

Find the square root of 3200. We can simplify \( \sqrt{3200} \) as follows:\( 3200 = 64 \times 50 \), thus \( \sqrt{3200} = \sqrt{64 \times 50} = 8 \times \sqrt{50} \). This further simplifies to approximately 56.57.

Key Concepts

The Role of Geometry in Problem Solving

Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, solids, and higher dimensional analogs. It is a vast field, but one of its simple and practical applications involves understanding shapes and their characteristics. In this exercise, we are dealing with a square, a fundamental 2-dimensional shape in geometry.

Squares, by definition, have equal sides and four right angles. Here, the mat forms a square with each side measuring 40 feet. The interest lies in finding the diagonal, which is a line segment connecting two opposite corners of the square. When you draw this diagonal, it naturally divides the square into two equal right triangles. The principles of geometry guide us to analyze the relationships between the sides of this square as the framework for solving the diagonal length problem.

• Understand and visualize shapes like squares.
• Acknowledge properties like equal side lengths and right angles.
• Recognize how a square relates to right triangles and their properties.

By using basic geometric principles, solving the diagonal becomes clearer and manageable, merging visual understanding with mathematical concepts.

Understanding Right Triangles and the Pythagorean Theorem

Right triangles are fundamental in trigonometry and geometry, known for having one angle measuring 90 degrees. These triangles are significant due to the powerful relation dictated by the Pythagorean Theorem.

The Pythagorean Theorem elegantly connects the sides of a right triangle. It states: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse, the side opposite the right angle.

In this problem, the diagonal of the square acts as this hypotenuse. Both legs (\( a \) and \( b \)) are identical, measuring 40 feet each. By applying the theorem, we calculate the diagonal by setting up the equation with these known values:

• The sides of the square become the two perpendicular sides of the triangle.
• The diagonal becomes the hypotenuse we're solving for.

Hence, using the Pythagorean Theorem helps us relate these measures to find the length of the diagonal in real-world applications.

Exploring Square Roots in Solving for Diagonals

Square roots play a crucial role when dealing with the Pythagorean Theorem, as they provide a means to solve for the hypotenuse when given the lengths of the two other sides. In our case, after setting up our equation with the known side lengths, we reached:\[ 3200 = c^2 \]

Finding \( c \), or the diagonal of our square mat, involves taking the square root of 3200.

• The calculation of \( \sqrt{3200} \) can first be simplified by factorizing 3200.
• Breaking down 3200 into \( 64 \times 50 \) simplifies it: \( \sqrt{3200} = \sqrt{64 \times 50} = 8 \times \sqrt{50} \).
• This further resolves to an approximate value: \( c \approx 56.57 \), where \( 8 \) is the square root of \( 64 \).

Thus, understanding how to compute square roots allows us to solve for distances and diagonals, which are pivotal in geometry and real-life scenarios. Such calculations emphasize the practical utility of mathematics.