Students are designing a new town as part of a social studies project on urban planning. ey want to place the town’s high school at point A and the middle school at point B. they also plan to build roads that run directly from point A to the mall and from point B to the mall. the average cost to build a road in this area is $550,000 per mile. a. Find the measure of each acute angle of the right triangle shown. b. Find the length of the hypotenuse. Also nd the length of each of the three congruent segments forming the hypotenuse.

The downtown angle measures about 22.62° and the town pool angle measures about 67.38°.

The hypotenuse measures 13 miles. Each third measures 4 1/3 miles.

Step-by-step explanation:

Use trigonometric ratios to solve this problem.

a. Find the measure of each acute angle of the right triangle shown.

Let's start with the acute angle that's marked near downtown. We can use the trigonometric ratio tangent to find the measure of this angle. (Remember, tangent = opposite/adjacent!)

We can divide the opposite side from the angle by the adjacent side to find the tangent:

5/12

= 0.4166666...

Now, we do the inverse tangent to find the measure of the angle:

≈ 22.62°

Now, we can find the angle near the town pool. This time, we can use tangent again, but the 12 mi side is the opposite and the 5 mi side is the adjacent:

12/5

= 2.4

Now, calculate the inverse tangent:

≈ 67.38°

b. Find the length of the hypotenuse. Also find the length of each of the three congruent segments forming the hypotenuse.

We can use the Pythagorean theorem to find the length of the hypotenuse. Remember, given legs a and b and hypotenuse c, the Pythagorean theorem states:

a² + b² = c²

Plug the values in this triangle into this equation:

5² + 12² = c²

25 + 144 = c²

169 = c²

13 = c

The hypotenuse measures 13 miles.

Now, we find the length of the three congruent segments that form the hypotenuse. (to be honest, I'm not sure that there are three congruent segments, but oh well, I'll just go with what it says there).

Since all the segments are the same length, we can just divide 13 by 3 to find the length of each of them:

13/3 = 4 1/3.

Each of the segments measures 4 1/3 miles.