a) Pine Road and Oak Street form a right angle, so we can extract the relation
[tex]\tan30^\circ=\dfrac x{11\sqrt3}\implies\dfrac1{\sqrt3}=\dfrac x{11\sqrt3}\implies x=11[/tex]
where [tex]x[/tex] is the distance we want to find (bottom side of the rectangle).
Alternatively, we can use the other given angle by solving for [tex]x[/tex] in
[tex]\tan60^\circ=\dfrac{11\sqrt3}x[/tex]
but we'll find the same solution either way.
b) Pine Road and Oak Street form a right triangle, with Main Street as its hypotenuse. We can use the Pythagorean theorem to find how long it is.
[tex](\text{Pine})^2+(\text{Oak})^2=(\text{Main})^2[/tex]
Let [tex]y[/tex] be the length of Main Street. Then
[tex](11\sqrt3)^2+11^2=x^2\implies x^2=484\implies x=\pm22[/tex]
but of course the distance has to be positive, so [tex]x=22[/tex].
