Example 6 from Algebra and Trigonometry, 10.2 Non-right Triangles: Law of Cosines

Applying Heron’s Formula to a Real-World Problem

A Chicago city developer wants to construct a building consisting of artist’s lofts on a triangular lot bordered by Rush Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square meters are available to the developer? See Figure 10 for a view of the city property.

A triangle formed by sides Rush Street, N. Wabash Ave, and E. Pearson Street with lengths 62.4, 43.5, and 34.1, respectively.

Figure 10

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Find the measurement for $s,$ which is one-half of the perimeter.

$\begin{array}{l} s = \frac{(62.4 + 43.5 + 34.1)}{2} \\ s = 70 m \end{array}$
Apply Heron’s formula.
$\begin{array}{l} Area = \sqrt{70 (70 - 62.4) (70 - 43.5) (70 - 34.1)} \\ Area = \sqrt{506,118.2} \\ Area \approx 711.4 \end{array}$
The developer has about 711.4 square meters.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the area of a triangle given $a = 4.38 ft, b = 3.79 ft,$ and $c = 5.22 ft .$

How did it go?

Solution

Your turn

Solution

Your turn