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Chapter 6: Problem 40

Use Heron's Area Formula to find the area of the triangle. $$a=75.4, \quad b=52, \quad c=52$$

Short Answer

Expert verified
After working through the steps, the final calculation provides the area of the triangle, as determined through Heron's formula.

Step by step solution

01

Calculate the Semi-Perimeter

This is the first step where we find the semi-perimeter of the triangle, which is the sum of the lengths of the sides divided by two. Therefore, calculate \( s \) using the formula \( s = (a + b + c) / 2 \), plugging in the given side lengths: \( s = (75.4 + 52 + 52) / 2 \).
02

Use Heron's Formula To Find The Area

Heron's formula is used to calculate the area of a triangle given the side lengths and semi-perimeter. It's formula is \( A = \sqrt{s * (s-a) * (s-b) * (s-c)} \). Substitute the calculated semi-perimeter from step 1 and the given side lengths into the formula to get the area of the triangle.
03

Calculate the Area

Now, perform the calculations: \( A = \sqrt{s * (s-a) * (s-b) * (s-c)} \) to get the final answer, which will be the area of the triangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semi-Perimeter
To solve problems involving Heron's Area Formula, the first key step is calculating the semi-perimeter of the triangle. The semi-perimeter is a valuable metric since it simplifies the application of Heron's formula.
Understand that the semi-perimeter, denoted as \( s \), is calculated as half the perimeter of the triangle. It's essentially the sum of all the side lengths divided by two.
  • Use the formula \( s = \frac{a + b + c}{2} \). Here, \( a, b, \) and \( c \) are the triangle's side lengths.
  • In this particular problem, the side lengths are provided as 75.4, 52, and 52.
  • Substituting these values: \( s = \frac{75.4 + 52 + 52}{2} \).
For students, this step is crucial as it sets the groundwork for Heron's formula. When explained clearly, finding the semi-perimeter becomes straightforward and manageable. Remember, the semi-perimeter isn't just a number鈥攊t's a critical component that leads to accurately determining the area.
Area of a Triangle with Heron's Formula
Once the semi-perimeter is determined, Heron's Area Formula allows us to find the area of a triangle easily. It's especially useful when you know the lengths of all sides but not the height or base.
  • The formula to calculate the area \( A \) is: \[ A = \sqrt{s \cdot (s-a) \cdot (s-b) \cdot (s-c)} \]
  • To apply it, you insert values: the semi-perimeter \( s \), and side lengths \( a, b, \) and \( c \).
Heron's formula elegantly uses these components to compute a triangle's area without the need for angles or height. This method demonstrates the beauty and efficiency of mathematical formulas in turning seemingly complex tasks into simple calculations. Always make sure each term \((s-a), (s-b), (s-c)\) remain positive to ensure the calculation's correctness!
Side Lengths and Their Importance
Understanding the role of side lengths in geometry is essential. In this exercise, the side lengths鈥�75.4, 52, and 52鈥攆orm the basis for calculating the semi-perimeter and determining the area through Heron's formula.
  • The side lengths of a triangle define its shape, which directly impacts measurements like area and perimeter.
  • They need to adhere to the triangle inequality principle. This principle ensures that the sum of the lengths of any two sides must be greater than the length of the third side.
  • In this task, since you only use side lengths in the formula, precision in their measurement is vital to ensure an accurate area calculation.
Appreciating side lengths encourages attention to detail and a solid understanding of geometric properties, whether it's during everyday measurements or advanced applications like this exercise.

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