The distance from jonestown to Maryville is 180 miles, the distance from maryville to elm city is 300 miles, and the distance from elm city to jonestown is 240 miles. do the three towns form a right triangle? explain

We can create a triangle where each town is a vertex. Let us call this triangle JEM (following the first letter of each town). The distance from one town to another is the length of a side of the triangle.

The distance from Jonestown to Maryville is 180 miles can be interpreted as JM = 180 miles.

Similarly we get:
JE = 240 miles
ME = 300 miles

If this were a right triangle it would satisfy the pythagorean theorem. The pythagorean theorem states that [tex] a^{2} + b^{2} = c^{2} [/tex] where c is the length of the hypotenise (the longest side) while a and b represent the length of the other two sides (called the legs of the triangle). So we take the distances above, plug them into the formula and see if the left side and right side are indeed equal.

ME is the longest side (300 miles) so we use 300 as c. We get:
[tex] 180^{2} + 240^{2} = 300^{2} [/tex]
[tex]32400+57600=90000[/tex]
[tex]90000=90000[/tex]

Therefore, the theorem is satisfied and the triangle is a right triangle.