Question

applying theorems to real - world scenarios
lucy has to run two errands. she starts from home and travels 3 miles south to the post office. from the post office, she travels 4 miles east to the gas station. then, from the gas station, she travels 5 miles to return home. the entire trip forms a triangle.
what was the smallest angle made in her trip?
it is at the gas station.
it is at lucys home.
it is at the post office.
it depends on the direction lucy is traveling.

Explanation:

Step1: Identify the triangle type

First, we recognize that the triangle formed has sides of length 3, 4, and 5 miles. Since \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\), this is a right - triangle, with the right angle at the post office (because the legs of the right - triangle are the 3 - mile (south) and 4 - mile (east) legs, which meet at the post office).

Step2: Recall the relationship between side lengths and angles in a triangle

In a triangle, the smallest angle is opposite the shortest side. The sides of the triangle are \(a = 3\) (opposite the angle at the gas station), \(b = 4\) (opposite the angle at Lucy's home), and \(c = 5\) (opposite the right angle at the post office). The shortest side is 3 miles, which is opposite the angle at the gas station. Wait, no, wait. Wait, in a right - triangle, the right angle is the largest angle. The other two angles are acute. Let's label the triangle: Let \(H\) be home, \(P\) be post office, \(G\) be gas station. So \(HP = 3\), \(PG = 4\), \(GH=5\). Angle at \(P\) is right angle (since south and east are perpendicular). Then, by the law of sines, \(\frac{\sin\angle H}{PG}=\frac{\sin\angle G}{HP}=\frac{\sin\angle P}{GH}\). \(\sin\angle H=\frac{PG\sin\angle P}{GH}=\frac{4\times1}{5} = 0.8\), so \(\angle H=\arcsin(0.8)\approx53.13^{\circ}\). \(\sin\angle G=\frac{HP\sin\angle P}{GH}=\frac{3\times1}{5}=0.6\), so \(\angle G = \arcsin(0.6)\approx36.87^{\circ}\). So the smallest angle is at the gas station? Wait, no, wait the options: the options are about where the smallest angle is. Wait, the angle at home: between \(GH = 5\) and \(HP = 3\). The angle at gas station: between \(GH = 5\) and \(PG = 4\). The angle at post office: right angle. So the smallest angle is at the gas station? Wait, no, wait my calculation: angle at \(G\) is \(\approx36.87^{\circ}\), angle at \(H\) is \(\approx53.13^{\circ}\), angle at \(P\) is \(90^{\circ}\). So the smallest angle is at the gas station. Wait, but let's re - check. The side opposite angle at \(G\) is \(HP = 3\), side opposite angle at \(H\) is \(PG = 4\), side opposite angle at \(P\) is \(GH = 5\). Since in a triangle, the smaller the side, the smaller the angle opposite to it. So the smallest side is 3 (opposite angle at \(G\) (gas station)), so the smallest angle is at the gas station. Wait, but the options: "It is at the gas station." is one option. Wait, but earlier I thought maybe I made a mistake. Wait, let's re - express:

Wait, the triangle has sides 3, 4, 5. Right - triangle at \(P\) (post office). So angles: at \(P\): \(90^{\circ}\), at \(H\): let's calculate using trigonometry. In triangle \(HPG\), right - angled at \(P\), \(\tan\angle H=\frac{PG}{HP}=\frac{4}{3}\), so \(\angle H=\arctan(\frac{4}{3})\approx53.13^{\circ}\). \(\tan\angle G=\frac{HP}{PG}=\frac{3}{4}\), so \(\angle G=\arctan(\frac{3}{4})\approx36.87^{\circ}\). So the smallest angle is \(\angle G\), which is at the gas station. So the answer is "It is at the gas station."

Answer:

It is at the gas station.