Question

test 4 right triangles and trigonometry

1. in fayetteville, the library is 3 miles due west of the post office and the zoo is 5 miles due north of the library. what is the distance from the post office to the zoo?
2. two vertical posts stand side by side. one post is 8 feet tall and the other is 17 feet tall. if a 24 - foot wire is stretched between the tops of the posts, how far apart are the posts?
3. a 15.6 - inch laptop is featured in a sales ad for $799. this distance is the diagonal distance across the screen. if the screen measures 7.8 inches in height, what is the width of the screen?

determine whether the triangle with the given side lengths is acute, right, obtuse, or not a triangle.

1. 3,9,10
2. 6,12,19
3. 21,28,35

Explanation:

Step1: Analyze problem 5

We can model the situation as a right - triangle. The distance from the post - office to the library is one leg \(a = 3\) miles, and the distance from the library to the zoo is the other leg \(b=5\) miles. We use the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).

$$c=\sqrt{3^{2}+5^{2}}=\sqrt{9 + 25}=\sqrt{34}\approx5.83$$

Step2: Analyze problem 6

The difference in height of the two posts is \(h=17 - 8=9\) feet. The wire length is the hypotenuse \(c = 24\) feet. Let the distance between the posts be \(x\). Using the Pythagorean theorem \(x=\sqrt{c^{2}-h^{2}}\)

$$x=\sqrt{24^{2}-9^{2}}=\sqrt{(24 + 9)(24 - 9)}=\sqrt{33\times15}=\sqrt{495}\approx22.25$$

Step3: Analyze problem 7

The diagonal of the laptop screen is the hypotenuse \(c = 15.6\) inches and the height is \(a = 7.8\) inches. Let the width be \(b\). Using the Pythagorean theorem \(b=\sqrt{c^{2}-a^{2}}\)

$$b=\sqrt{15.6^{2}-7.8^{2}}=\sqrt{(15.6 + 7.8)(15.6-7.8)}=\sqrt{23.4\times7.8}=\sqrt{182.52}\approx13.51$$

Answer:

1. Approximately \(5.83\) miles
2. Approximately \(22.25\) feet
3. Approximately \(13.51\) inches