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Chapter 5: Problem 40

Use Heron's Area Formula to find the area of the triangle. $$a=\frac{3}{5}, \quad b=\frac{5}{8}, \quad c=\frac{3}{8}$$

Short Answer

Expert verified
Therefore, based on Heron's formula, the triangle area is \(\frac{1}{40}\)

Step by step solution

01

Calculating the Semiperimeter of the Triangle

First, calculate the semiperimeter of the triangle. The semiperimeter is half the perimeter and can be found using the formula \(s = \frac{a+b+c}{2}\). Given \(a=\frac{3}{5}\), \(b=\frac{5}{8}\), and \(c=\frac{3}{8}\), this becomes \(s = \frac{\frac{3}{5} + \frac{5}{8} + \frac{3}{8}}{2} = \frac{1}{2}\).
02

Applying Heron's Formula

Next, apply the Heron's formula that is \(A = \sqrt{s(s-a)(s-b)(s-c)}\). Substituting \(s = \frac{1}{2}\), \(a=\frac{3}{5}\), \(b=\frac{5}{8}\), and \(c=\frac{3}{8}\) into the equation to get the area, it becomes \(A = \sqrt{\frac{1}{2}(\frac{1}{2}-\frac{3}{5})(\frac{1}{2}-\frac{5}{8})(\frac{1}{2}-\frac{3}{8})}\)
03

Solving for the Area

Solve the expression in the square root to get the area of the triangle. The result is \(A = \sqrt{\frac{1}{2} * (-\frac{1}{10}) * (-\frac{1}{8}) * (\frac{1}{8})} = \frac{1}{40}\)

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