Problem 31 - Algebra 1: Concepts and Skills

Question

The length of each side of a baseball diamond is 90 feet. What is the diagonal distance \(c\) from home plate to second base?

Step-by-Step Solution

Answer

The diagonal distance from home plate to second base is approximately 127.28 feet.

Step 1: Identify the values of sides a and b

In this case, each side of the baseball diamond, which is a square, is 90 feet. Therefore, side \(a = 90\) feet and side \(b = 90\) feet.

Step 2: Apply the Pythagorean theorem

To find the length of the hypotenuse or the diagonal from home plate to second base, we apply the Pythagorean theorem. The formula is \(c = \sqrt{a^2 + b^2}\). By substituting the known values, we have \(c = \sqrt{90^2 + 90^2}\).

Step 3: Calculate the value of c

Now we solve the equation \(c = \sqrt{90^2 + 90^2}\) to find the diagonal length. After calculation, the answer is \(c \approx 127.28\) feet

Key Concepts

Diagonal of a Square

In the context of a baseball diamond, understanding the diagonal of a square is essential for calculating distances across the field. A square is a special kind of rectangle where all four sides are of equal length. To find the diagonal of a square, you can use the Pythagorean Theorem.
When you are given a square with side length of 90 feet, like that of a baseball field, the diagram forms two identical right triangles across the diagonal.

• The diagonal acts as the hypotenuse of these triangles.
• Both legs of the triangles are equal to the side length of the square, which is 90 feet in this case.

Because all sides are equal, calculating the diagonal becomes straightforward with the Pythagorean Theorem, where the diagonal is the hypotenuse of a right triangle formed in the square.

Right Triangle

The right triangle emerges when you draw a diagonal in a square. Understanding
its properties aids in applying the Pythagorean Theorem correctly. A right triangle
has one angle that is exactly 90 degrees, known as the right angle. The side opposite
this right angle is known as the hypotenuse, which is the longest side.
In our square baseball diamond, the diagonal serves as the hypotenuse:

• The two equal sides of the triangle, each 90 feet, form the two other angles of the triangle.
• This symmetry allows you to directly apply formulas like the Pythagorean Theorem efficiently.

By deriving these properties, you understand why calculating the diagonal helps find crucial distances like from home plate to second base using simple geometry.

Geometry Applications

Geometry is highly useful in calculating distances and designing spaces in real-world applications like sports fields. Knowing how to apply geometrical principles, such as the Pythagorean Theorem, allows us to solve practical problems.
The baseball diamond is a great example:

• Each corner forms a square, making it easy to partition it into right triangles and find distances.
• By understanding squares and right triangles, players and coaches can make strategic decisions based on field dimensions.

This understanding not only aids in planning game strategies but also enhances the comprehension of how mathematical concepts are relevant in various contexts.