Question

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 Law of Cosines

For a triangle with sides $a,b$ and included angle $\theta$, the third side $c$ is: $c^2 = a^2 + b^2 - 2ab\cos\theta$. Here $a=b=20$ ft, so $c = \sqrt{20^2 + 20^2 - 2(20)(20)\cos\theta} = 20\sqrt{2(1-\cos\theta)}$.

Step2: Relate angle to side length

$\cos\theta$ decreases as $\theta$ increases from $0^\circ$ to $180^\circ$, so $1-\cos\theta$ increases. Thus, larger $\theta$ → longer $c$ (fence).

Step3: Compare angles

Lot1: $79^\circ$, Lot2: $65^\circ$, Lot3: $36^\circ$. Order of angles: $79^\circ > 65^\circ > 36^\circ$.

Answer:

Longest fence: Lot 1
Shortest fence: Lot 3