Question

applying the hinge theorem to real - world problems
lot 1 lies where two roads meet at a 79° angle.
lot 2 lies where two roads meet at a 65° angle.
lot 3 lies where two roads meet at a 36° angle.
all lots have two sides measuring 20 feet as shown.
a fence runs along the back wall of each property, as shown in the diagram.
which lot has the longest fence along the back wall?
which lot has the shortest fence along the back wall?

Explanation:

Step1: Recall the Hinge Theorem

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

Step2: Compare the angles

We have three lots. For Lot 1, the included angle \(\theta_1 = 79^{\circ}\), for Lot 2, \(\theta_2=65^{\circ}\), and for Lot 3, \(\theta_3 = 36^{\circ}\). Since \(79^{\circ}>65^{\circ}>36^{\circ}\) and the two - side lengths (20 feet) of each lot are equal.

Step3: Apply the Hinge Theorem

By the Hinge Theorem, the side (fence) opposite the largest included angle is the longest and the side opposite the smallest included angle is the shortest.

Answer:

The longest fence is for Lot 1. The shortest fence is for Lot 3.