Question

Due to construction, traffic is getting detoured from Main Street by turning slightly right onto Oak Avenue and continuing straight for 4 mi. To get back onto Main Street, you need to make a 90 degree left turn onto Lilac Lane and continue straight for x miles. Lilac Lane intersects Main Street at 30 degrees for the end of the detour. Approximately how many miles should you travel on Lilac Lane to reach Main Street?

Answer 1

For this case you should see the problem as a rectangle triangle whose sides are 4 miles (base) and x miles (height).
 the angle between the hypotenuse and the base is 30 degrees.
 then to find the height:
 tan (30) = x / 4
 x = 4 * tan (30) = 2.31
 answer
 you should travel 2.31 miles on Lilac Lane to reach Main Street

Answer 2

6.9 miles
   Assuming that Main Street, Oak Avenue, and Lilac Lane are all straight, you have a right 30/60/90 right triangle. I do object to the phrase "turning slightly right onto Oak" since I wouldn't consider a 60 degree turn to be "slight". But in any case, the short leg of the 30/60/90 triangle is the 4 miles given and the long leg will be sqrt(3) times longer. So
 sqrt(3) * 4 miles = 1.732050808 * 4 miles = 6.92820323 miles
   So you would need to travel about 6.9 miles on Lilac Lane to reach Main Street.