Problem 57 - Precalculus with Limits

Question

Geometry The lengths of the sides of a triangular parcel of land are approximately 200 feet, 500 feet, and 600 feet. Approximate the area of the parcel.

Step-by-Step Solution

Answer

The area of the parcel is approximately 50,000 square feet.

Step 1: Identify the Base and the Height

As the triangle is a right triangle, the longest side will be the hypotenuse. The other two sides can be considered as base and height. Thus, the base would be 500 feet and the height would be 200 feet.

Step 2: Use the Formula for Area of a Triangle

Here, the formula for area of a triangle 1/2(base * height) can be used. Substituting the values we get area as 1/2(500 * 200).

Step 3: Calculate Area

Now, calculate the area as per the formula. The result is 50,000 square feet.

Key Concepts

Triangles

Triangles are fascinating shapes in geometry. They are polygons with three edges and three vertices. In any triangle, the sum of the internal angles is always 180 degrees. This fundamental property makes them unique and useful in various applications ranging from architecture to calculating areas of parcels of land.

There are different types of triangles, which are usually classified by their sides or angles:

• Equilateral Triangle: All three sides are equal, and each angle measures 60 degrees.
• Isosceles Triangle: Has two sides of equal length, with two equal angles.
• Scalene Triangle: All sides and angles are different.
• Right Triangle: Has one 90-degree angle, making it particularly important in geometry.

Understanding the properties of these triangles lays the groundwork for solving problems related to their area, perimeter, and more.

Area Calculation

Calculating the area of a triangle involves understanding its base and height. The area of a triangle is given by the formula:\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]This formula is derived from the geometry of the triangle itself, where the base is any of its sides, and the height is the perpendicular distance from the base to the opposite vertex.

For example:

• If a triangle has a base of 500 feet and a height of 200 feet, its area can be calculated by substituting these values into the formula.
• This equation simplifies the often complex nature of determining area, particularly in right triangles where the base and height are directly measurable from the sides.

Using this straightforward formula efficiently gives us the area, which in this case is 50,000 square feet, helping us to solve real-world problems such as determining the usable space of a land parcel.

Right Triangles

Right triangles are a special category of triangles featuring one 90-degree angle. This right angle not only defines the triangle but also leads to some of the most pivotal geometric principles, such as the Pythagorean theorem.

In a right triangle:

• The longest side opposite the right angle is known as the hypotenuse.
• The other two sides are referred to as the legs, and these form the basis and height for area calculations.

The Pythagorean theorem states that the square of the hypotenuse (\( c \)) is equal to the sum of the squares of the other two sides (\( a \) and \( b \)), given by:\[c^2 = a^2 + b^2\]
Right triangles are prevalent in many practical scenarios, including construction and navigation, due to their straightforward properties and easy calculations. This makes them an essential section of geometry, especially when analyzing real-world applications like the area of land as in this exercise.