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 the post office.
it is at lucy’s home.
it depends on the direction lucy is traveling.

Explanation:

Step1: Identify the triangle sides

Lucy's path forms a triangle with sides \( a = 3 \) (home to post office), \( b = 4 \) (post office to gas station), and \( c = 5 \) (gas station to home). This is a right - triangle since \( 3^{2}+4^{2}=9 + 16=25 = 5^{2}\) (Pythagorean theorem).

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

In a triangle, the smallest angle is opposite the shortest side. The shortest side here is \( a = 3 \) (home to post office), and the angle opposite to this side is the angle at the gas station (because the side of length 3 is between home and post office, so the angle opposite is at the gas station, formed by the sides of length 4 and 5).

Step3: Confirm the angle location

Using the property that in a triangle, smaller sides are opposite smaller angles. The side of length 3 (home - post office) is opposite the angle at the gas station. So the smallest angle is at the gas station.

Answer:

It is at the gas station.