Problem 32 - Algebra 1: Concepts and Skills

Question

DIAGONAL OF A FIELD A field hockey field is a rectangle 60 yards by 100 yards. What is the length of the diagonal from one corner of the field to the opposite corner?

Step-by-Step Solution

Answer

The diagonal of the field is approximately 116.62 yards.

Step 1: Understand the Problem

Take note of the dimensions of the field which are 60 yards and 100 yards. Identify that you are required to find the length of the diagonal which forms the hypotenuse of the right angle triangle.

Step 2: Apply the Pythagorean Theorem

By the Pythagorean Theorem, the formula for the diagonal \(d\) is \(d = \sqrt{width^2 + length^2}\). So substitute the given values into the formula, \(d = \sqrt{60^2 + 100^2}\).

Step 3: Compute the Diagonal

Perform the calculation: \(d = \sqrt{3600 + 10000} = \sqrt{13600}\).

Step 4: Simplify the Expression

Simplify the square root, this yields the length of the diagonal: \(d = 116.62\) yards.

Key Concepts

Understanding a Right Triangle

A right triangle is a type of triangle where one of its angles measures exactly 90 degrees. This 90-degree angle is known as a right angle. In any right triangle, the side opposite this right angle is called the hypotenuse. This side is always the longest because it stretches directly from one corner of the right angle to the other corner.
The other two sides, which form the right angle, are known as the legs of the triangle.

• One leg is called the "base," and the other is the "height."
• The right angle forms a perfect L-shape when looking at the diagram of the triangle.

Understanding these basic properties of a right triangle helps solve problems involving distances, such as finding the diagonal of a rectangular field.

The Importance of the Hypotenuse

The hypotenuse plays a crucial role in solving problems involving right triangles, especially when applying the Pythagorean Theorem. It represents the diagonal in the scenario of a rectangular field problem. Knowing how to calculate the hypotenuse aids in determining unknown distances.

• The Pythagorean Theorem states that in a right triangle: \[ c^2 = a^2 + b^2 \]
• In this equation, \( c \) signifies the hypotenuse, whereas \( a \) and \( b \) are the legs of the triangle.

The calculation of the hypotenuse using this theorem is essential for engineers, architects, and even athletes who need to understand diagonal distances.

Exploring Geometry Concepts

Geometry is the area of mathematics concerned with shapes, sizes, and the properties of space. In the context of this exercise, geometry helps us understand the relationships between different elements of shapes such as rectangles and triangles. When we calculate the diagonal of a rectangle using the Pythagorean Theorem, we're using geometry principles.

• Geometry involves understanding angles, lengths, and the interplay between various shapes.
• Rectangles have four right angles, and when divided by a diagonal, they form two congruent right triangles.

Practicing these concepts helps us solve practical problems in science, architecture, and everyday life.

Significance of a Rectangular Field

A rectangular field is a quadrilateral with four right angles, and it is common in sports such as soccer and field hockey. Understanding its properties is crucial for tasks related to sports field measurements and design.

• Each side opposite to one another is of equal length in a rectangle.
• The diagonal divides the rectangle into two equal right triangles.
• Measuring the diagonal helps in setting up fields correctly and optimizing layouts.

Applying these concepts allows groundskeepers to measure diagonals accurately, ensuring each half of the field is equal for fair play.