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 (also known as the SAS Inequality 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. Conversely, if the included angle is smaller, the third side is shorter.

Step2: Analyze the given lots

All three lots (Lot 1, Lot 2, Lot 3) have two sides of length 20 feet (so the two sides are congruent for each triangle). The included angles are:

• Lot 1: \(79^\circ\)
• Lot 2: \(65^\circ\)
• Lot 3: \(36^\circ\)

Step3: Compare the included angles for the longest fence

To find which lot has the longest fence (third side), we need the largest included angle. Among \(79^\circ\), \(65^\circ\), and \(36^\circ\), \(79^\circ\) is the largest. So Lot 1, with the largest included angle, will have the longest third side (fence).

Step4: Compare the included angles for the shortest fence

To find which lot has the shortest fence (third side), we need the smallest included angle. Among \(79^\circ\), \(65^\circ\), and \(36^\circ\), \(36^\circ\) is the smallest. So Lot 3, with the smallest included angle, will have the shortest third side (fence).

Answer:

Longest fence: Lot 1
Shortest fence: Lot 3