Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

The lengths of the sides of a triangular garden at a university are approximately 160 feet, 150 feet, and 140 feet. Approximate the area of the garden.

Step-by-Step Solution

Answer

The area of the garden is approximately 7572 square feet.

Step 1: Find the Semiperimeter

First, calculate the semiperimeter (s) of the triangle. This is given by the formula \(s = \frac{a + b + c}{2}\), where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle. Substituting the given values, we get \(s = \frac{160 + 150 + 140}{2} = 225\).

Step 2: Apply Heron's Formula

Next, substitute these values into Heron's formula: \(Area = \sqrt{s(s-a)(s-b)(s-c)}\). Substituting, we get \(Area = \sqrt{225(225-160)(225-150)(225-140)}\). Calculating the terms inside the square root gives us a value inside the root of 57337500.

Step 3: Find the Square Root to get the Area

Lastly, find the square root of 57337500 to get the area of the garden. This gives an area of approximately 7572 square feet.

Key Concepts

Triangle Area Calculation

When it comes to calculating the area of a triangle, there are several methods. One of the most powerful and efficient is Heron's Formula, especially when you know the lengths of all three sides. This formula allows you to find the area without needing to know the height, which can be complex to measure in real-world scenarios.Heron's Formula is given by:\[ Area = \sqrt{s(s-a)(s-b)(s-c)} \]Here, \(s\) is the semiperimeter, and \(a\), \(b\), and \(c\) are the side lengths of the triangle. This formula is particularly useful because it encompasses all sides of the triangle, providing a comprehensive way to determine its area. This enables architects, surveyors, and garden planners to design and adjust spaces efficiently.

Semiperimeter in Triangles

The semiperimeter of a triangle is a crucial stepping stone in applying Heron's Formula. It's defined as half the triangle's perimeter and is expressed as follows:\[ s = \frac{a + b + c}{2} \]Understanding the role of the semiperimeter helps streamline the area calculation process. First, you add up all the side lengths of the triangle and then divide the sum by two. For example, in the original exercise, the sides were 160 feet, 150 feet, and 140 feet:

• Add up the side lengths: 160 + 150 + 140 = 450 feet
• Divide by 2 to get the semiperimeter: 450 / 2 = 225 feet

Thus, the semiperimeter equips you with a pivotal number needed to apply Heron's formula correctly.

Geometry Problem Solving

Geometry is not just about knowing formulas; it’s about understanding the concepts that drive those formulas. Solving a geometry problem often involves several steps, each building on the previous one. Let's break it down further using our triangular garden problem:

• **Identify what you need**: Here, knowing the lengths of all triangle sides and the area is required.
• **Calculate the semiperimeter**: This is crucial as it simplifies the process of applying Heron’s Formula.
• **Use Heron’s Formula**: By plugging in the semiperimeter and the side lengths, you can calculate the area.
• **Verify results**: Double-check calculations to ensure accuracy, especially when dealing with real-world applications.

These steps help break down complex problems and make it easier to tackle them systematically. Understanding each part of the solution gives you confidence and the foundational skills necessary to solve other geometric problems efficiently in the future.