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Chapter 7: Problem 31

Find the area of each triangle using Heron's formula. $$a=346, b=234, c=422$$

Short Answer

Expert verified
Approx. 40,548.66 square units.

Step by step solution

01

Calculate the Semiperimeter

The semiperimeter of a triangle is calculated by taking half of the sum of its three sides. Using the formula \(s = \frac{a+b+c}{2}\), substitute the given side lengths: \[s = \frac{346+234+422}{2} = 501\]
02

Apply Heron's Formula

Heron's formula for the area of a triangle is \(A = \sqrt{s(s-a)(s-b)(s-c)}\). Using the semiperimeter calculated in Step 1 (\(s = 501\)), and the given side lengths (\(a = 346\), \(b = 234\), \(c = 422\)), we have: \[A = \sqrt{501(501-346)(501-234)(501-422)}\]
03

Simplify Inside the Square Root

First, calculate the values inside the square root individually: \[501 - 346 = 155\]\[501 - 234 = 267\]\[501 - 422 = 79\]Next, substitute these values back into the square root expression: \[A = \sqrt{501 \times 155 \times 267 \times 79}\]
04

Calculate the Final Area

Compute the product inside the square root and then take the square root: \[501 \times 155 \times 267 \times 79 \approx 1,644,977,905\]Taking the square root, \[A = \sqrt{1,644,977,905} \approx 40,548.66\]

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Key Concepts

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

Area of a Triangle
The area of a triangle can be calculated using several methods. One common way is using Heron's formula when the lengths of all three sides are known. Heron's formula provides a direct way to calculate the area without knowing the height. This is extremely useful in cases where calculating or measuring the height is difficult.

To use Heron's formula, you first need to determine the semiperimeter, denoted as \(s\). You then substitute the semiperimeter and the side lengths into the formula: \[A = \sqrt{s(s - a)(s - b)(s - c)}\]

Let's walk through an example problem where \(a = 346\), \(b = 234\), and \(c = 422\).
Semiperimeter
The semiperimeter of a triangle is half the sum of its side lengths. It's a critical step in using Heron's formula. To calculate the semiperimeter \(s\) for our example, use the formula:

\[s = \frac{a + b + c}{2}\]

Substituting the given side lengths:

\[s = \frac{346 + 234 + 422}{2} = 501\]

The semiperimeter \(s = 501\) units. This value will now be used to find the area of the triangle in Heron's formula.
Triangle Side Lengths
The side lengths of a triangle are fundamental to many of its properties, including its area. In Heron's formula, you need all three side lengths.

For the example triangle with sides \(a = 346\), \(b = 234\), and \(c = 422\):
  • First, calculate the semiperimeter \(s = 501\).
  • Next, determine the individual terms in parentheses:

    \[s - a = 501 - 346 = 155\]
    \[s - b = 501 - 234 = 267\]
    \[s - c = 501 - 422 = 79\]


With all these values, substitute them back into Heron鈥檚 formula: \[A = \sqrt{501 \times 155 \times 267 \times 79}\]

Finally, calculate the resulting product under the square root and then take the square root:

\(\text{Area} \approx 40,548.66\) square units.
This process can be applied to any triangle as long as all three side lengths are known.

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