Problem 5 - Algebra 2

Question

In Australian baseball, the bases lie at the vertices of a square 27.5 meters on a side and the pitcher’s mound is 18 meters from home plate. Find the distance from the pitcher’s mound to first base.

Step-by-Step Solution

Answer

The distance is approximately 26.52 meters.

Step 1: Identify the Problem

We are asked to find the distance from the pitcher's mound to first base in a square-shaped baseball field. The problem involves using geometric properties of a square.

Step 2: Identify Known Values

The side of the square, representing the distance between bases, is given as 27.5 meters, and the distance from home plate to the pitcher's mound is 18 meters.

Step 3: Draw a Diagram

Visualize the baseball field by sketching a square with side length 27.5 meters. Place the pitcher's mound and the bases on the diagram, noting the given distances.

Step 4: Apply the Pythagorean Theorem

Use the Pythagorean Theorem because the problem involves finding the hypotenuse of a right-angled triangle, where the side from home to the pitcher is one leg, and the side from the pitcher to first base, being the unknown, is the hypotenuse. The formula is: \[ c = \sqrt{a^2 + b^2} \] where \( a \) is the distance from the pitcher's mound to home plate and \( b \) is half the diagonal of the square because the mound lies mid-way in the square's diagonal, which makes both legs equal at the midpoint.

Step 5: Calculate the Diagonal of the Square

Find the total diagonal of the square using the formula: \[ \text{Diagonal} = \sqrt{2} \times \text{Side Length} = \sqrt{2} \times 27.5 \approx 38.89 \text{ meters}. \]Since the pitcher's mound is at mid-point, each half is: \( b = 38.89 / 2 = 19.445 \text{ meters} \)

Step 6: Calculate the Pitcher's Distance to First Base

Substitute the values into the Pythagorean Theorem to find the hypotenuse (distance from pitcher's mound to first base): \[ c = \sqrt{18^2 + 19.445^2} \approx \sqrt{324 + 378.92} \approx \sqrt{702.92} \approx 26.52 \text{ meters}. \]

Step 7: Final Step: Record the Answer

The distance from the pitcher's mound to first base is approximately 26.52 meters.

Key Concepts

Geometric Properties of a Square

A square is a four-sided polygon, known as a quadrilateral, that has special geometric properties. Each side of a square is equal in length, and all of its internal angles are right angles, measuring 90 degrees. This symmetry gives the square unique characteristics that are useful when solving various geometry-related problems.

• All sides being equal means that if one side is known, the perimeter can be easily calculated by multiplying the side length by four.
• The diagonals of a square are equal in length and bisect each other at right angles, dividing the square into two congruent right-angled triangles.

Understanding these properties is crucial in tasks involving squares, such as determining the position of points relative to the square's structure. For instance, in the baseball field scenario, knowing the length of the sides (27.5 meters) allows us to easily calculate the length of the diagonal when needed.

Distance Calculation

Finding distances in geometric shapes often requires specific formulas and understanding of the spatial arrangement. In this exercise, we are tasked with calculating the distance between the pitcher's mound and first base, using the Pythagorean Theorem.

• First, the diagonal of the square is calculated using the formula \[ \text{Diagonal} = \sqrt{2} \times \text{Side Length} \]which gives us a key length dividing the square into two triangles.
• Knowing the pitcher's mound is midway along one diagonal portion, we calculate half this diagonal for one leg of a right-angled triangle.
• By using the Pythagorean Theorem, \[ c = \sqrt{a^2 + b^2} \],each known value is substituted to find the unknown length which is the hypotenuse, or the required distance.

Right-Angled Triangle

A right-angled triangle is a triangle where one angle measures 90 degrees. These triangles are foundational in geometry, especially because they allow the use of the Pythagorean Theorem. This theorem relates the lengths of the sides in a simple equation valuable for calculations.

• In a right-angled triangle, the side opposite the right angle is the hypotenuse, known as the longest side.
• The remaining sides, often referred to as the legs, are perpendicular to each other.
• The relationship between the sides is given by: \[ a^2 + b^2 = c^2 \], where "c" is the hypotenuse, while "a" and "b" are the legs.

In our exercise, once the right-angled triangle was established by the base paths of the baseball field, we utilized this triangle property to calculate necessary distances from a given midpoint to a base, assuring all computations were precise and consistent with geometric principles.