Question

As the city planner of Right City, you are responsible to report information to help the Board of Supervisors make decisions about the budgets for the fire and police departments. The board has asked for a report with answers to the following questions. Each grid unit in the figure at right represents
1
mile.
For fire safety, bushes will be cleared along the perimeter of the city. What is the length of the perimeter? Include all of your calculations in your report.
Some people say that Right City got its name from its shape. Is the shape of the city a right triangle? Show how you can tell.

Answer 1

The perimeter of the city is the sum of the triangle's sides. The ritgtmost side length, according to the coordinates of the side's ends, is

$$R=13-7=6 \text{ miles}.$$

The topmost side length is

$$T=\sqrt{(13-11)^2+(4-1)^2}=\sqrt{13} \text{ miles}.$$

The bottom side length

$$B=\sqrt{(11-7)^2+(4-1)^2}=5 \text{ miles}.$$

The perimeter is

$$P=R+T+B=(11+\sqrt{13})\text{ miles}.$$

If the city is right indeed, we must get

$$R^2=T^2+B^2$$

Check it!

$$6^2<\sqrt{13}^2+5^2.$$

This is not Right City.