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.
diagram of triangular lots with fences and angle labels
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

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

All lots have two sides of length \(20\) ft (congruent sides).

Step3: Determine the longest back fence

To find the lot with the longest back fence, we need the largest included angle (since larger included angle means longer third side by Hinge Theorem). Among \(79^\circ\), \(65^\circ\), and \(36^\circ\), \(79^\circ\) is the largest. So Lot 1 has the largest included angle, hence the longest back fence.

Step4: Determine the shortest back fence

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

Answer:

• Which lot has the longest fence along the back wall? Lot 1
• Which lot has the shortest fence along the back wall? Lot 3