Question

what is the approximate length of the missing side in the triangle below? triangle with sides 15 miles, 18 miles, and included angle 80° not drawn to scale options: 13.9 mi., 19.0 mi., 21.3 mi., 25.4 mi.

Explanation:

Step1: Identify the Law to Use

We have two sides of a triangle (15 mi, 18 mi) and the included angle (80°), so we use the Law of Cosines. The Law of Cosines formula for a triangle with sides \(a\), \(b\), and included angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). Let \(a = 15\), \(b = 18\), and \(C = 80^{\circ}\), and we need to find \(c\).

Step2: Substitute Values into the Formula

Substitute \(a = 15\), \(b = 18\), and \(C = 80^{\circ}\) into the formula:
\[
c^{2}=15^{2}+18^{2}-2\times15\times18\times\cos(80^{\circ})
\]
First, calculate \(15^{2}=225\), \(18^{2}=324\), so \(15^{2}+18^{2}=225 + 324=549\). Then, calculate \(2\times15\times18 = 540\). Now, find \(\cos(80^{\circ})\approx0.1736\). So, \(2\times15\times18\times\cos(80^{\circ})\approx540\times0.1736\approx93.744\).

Step3: Calculate \(c^{2}\)

Now, \(c^{2}=549 - 93.744=455.256\).

Step4: Find \(c\)

Take the square root of \(c^{2}\) to find \(c\): \(c=\sqrt{455.256}\approx21.3\) (rounded to one decimal place).

Answer:

21.3 mi.