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Chapter 7: Problem 36

Find the area (in square units) of each triangle described. $$ a=\frac{25}{6}, b=\frac{17}{6}, c=\frac{11}{6} $$

Short Answer

Expert verified
The area of the triangle is approximately 2.118 square units.

Step by step solution

01

Identify the Formula

To find the area of a triangle given its three sides, we use Heron's formula. Heron's formula states that for a triangle with sides of length \(a\), \(b\), and \(c\), the area \(A\) is given by: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \(s\) (the semi-perimeter) is calculated as \(s = \frac{a+b+c}{2}\).
02

Calculate Semi-Perimeter

Using the side lengths \(a = \frac{25}{6}\), \(b = \frac{17}{6}\), and \(c = \frac{11}{6}\), calculate \(s\) as follows: \[ s = \frac{\frac{25}{6} + \frac{17}{6} + \frac{11}{6}}{2} = \frac{\frac{53}{6}}{2} = \frac{53}{12} \]
03

Calculate Each Term of the Formula

Using the calculated semi-perimeter \(s = \frac{53}{12}\), calculate the individual components:- \(s-a = \frac{53}{12} - \frac{25}{6} = \frac{53}{12} - \frac{50}{12} = \frac{3}{12} = \frac{1}{4} \)- \(s-b = \frac{53}{12} - \frac{17}{6} = \frac{53}{12} - \frac{34}{12} = \frac{19}{12} \) - \(s-c = \frac{53}{12} - \frac{11}{6} = \frac{53}{12} - \frac{22}{12} = \frac{31}{12} \)
04

Plug into Heron's Formula

Substitute the values into Heron's formula to find the area:\[ A = \sqrt{\frac{53}{12} \times \frac{1}{4} \times \frac{19}{12} \times \frac{31}{12}} \] Simplify step-by-step to calculate the product:1. \( \frac{53 \times 1 \times 19 \times 31}{12 \times 4 \times 12 \times 12} = \frac{31103}{6912} \)2. \( A = \sqrt{\frac{31103}{6912}} \) The exact value is complicated, so you can find a numerical approximation of this square root.
05

Simplify and Approximate

Let's find the approximate value for better understanding: \[ A \approx \sqrt{\frac{31103}{6912}} \approx 2.118 \] Therefore, the approximate area of the triangle is around 2.118 square units.

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Key Concepts

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

Triangle Area Calculation
Calculating the area of a triangle can be easy if you know the lengths of its sides. One efficient way to determine a triangle's area is using Heron's formula. This formula is especially beneficial when you have a triangle with sides but no height given.
Heron's formula as given by:
  • First, find the semi-perimeter, denoted as \( s \), of the triangle, which is half of the perimeter.
  • Once you've got \( s \), Heron's formula is \( A = \sqrt{s(s-a)(s-b)(s-c)} \).
The beauty of Heron's formula lies in its straightforwardness. With just the side lengths, you can dive into calculating the area without needing the additional headache of finding the height or an extra angle. So, if you have the sides, go ahead and use Heron's formula for a quick area calculation.
Semi-Perimeter Calculation
The semi-perimeter of a triangle is a stepping stone to finding its area using Heron's formula. The term "semi" refers to the fact that it involves taking half of the triangle's perimeter. Here's how to find it:
  • First, add the lengths of the sides: \( a + b + c \).
  • Then, divide by 2: \( s = \frac{a+b+c}{2} \).
In the example given, the sides \( a = \frac{25}{6} \), \( b = \frac{17}{6} \), and \( c = \frac{11}{6} \) were used to calculate the semi-perimeter as \( s = \frac{53}{12} \).
Calculating the semi-perimeter is a critical step, as it leads directly into the application of Heron's formula, setting the stage for calculating the triangle's area.
Square Root Approximation
Once the semi-perimeter \( s \) is known, and all calculations of \( (s-a)(s-b)(s-c) \) are complete, plugging these into Heron's formula gives a long expression under the square root. Solving this part often requires a square root approximation.
  • First, evaluate the expression inside the square root: \( \sqrt{s(s-a)(s-b)(s-c)} \).
  • For numerical results, calculate the square root using a calculator to simplify \( A \).
In the example, \( A = \sqrt{\frac{31103}{6912}} \) simplifies to an approximate value of \( 2.118 \).
Calculating area using approximations makes the answer more tangible, helping to visualize the triangle's area in a real-world context.
Sides of a Triangle
Understanding the significance of a triangle's sides is crucial when applying formulas like Heron's. The lengths of sides play a decisive role in determining the shape and area of a triangle.
A triangle is defined by:
  • Three sides, typically denoted as \( a \), \( b \), and \( c \).
  • Heron's formula applies when all these side lengths are known, requiring no additional information about other triangle features like angles.
Calculating area with specific side lengths provides a consistent method to determine size. In this example, knowing the values \( a = \frac{25}{6} \), \( b = \frac{17}{6} \), and \( c = \frac{11}{6} \) served as a foundation to use Heron's formula for finding the triangle's area effectively. Recognizing these lengths signifies the triangle's character, from which its semi-perimeter and, subsequently, its area are derived.

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