Question

pretest: unit 4 given the triangle below, what is the length of the third side, rounded to the nearest whole number?
triangle with vertices a, b, c; side ab = 19, side ac = 12, angle at a is 56°; note: triangle not drawn to scale
options:
○ a. 17
○ b. 13
○ c. 16

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides and the included angle, so we use the Law of Cosines. The Law of Cosines formula for a triangle with sides \(a\), \(b\), \(c\) and included angle \(C\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). Here, let \(a = 19\), \(b = 12\), and the included angle \(A=56^{\circ}\), and we want to find side \(BC\) (let's call it \(c\)). So the formula becomes \(c^{2}=19^{2}+12^{2}-2\times19\times12\times\cos(56^{\circ})\).

Step2: Calculate Each Term

First, calculate \(19^{2}=361\), \(12^{2} = 144\). Then, calculate \(2\times19\times12=456\). Now, find \(\cos(56^{\circ})\approx0.5592\). Then, \(2\times19\times12\times\cos(56^{\circ})\approx456\times0.5592\approx254.9952\).

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

Now, \(c^{2}=361 + 144-254.9952=505 - 254.9952 = 250.0048\).

Step4: Find \(c\)

Take the square root of \(c^{2}\): \(c=\sqrt{250.0048}\approx15.81\), which rounds to 16.

Answer:

C. 16