Question

lee is creating a garden with the dimensions shown. he creates paths through the garden along $overline{ab}$, $overline{xy}$, and $overline{zy}$. which correctly compares the lengths of the paths? the path along $overline{xy}$ is longer than the path along $overline{zy}$, and the path along $overline{zy}$ is longer than the path along $overline{ab}$. the path along $overline{ab}$ is longer than the path along $overline{zy}$, and the path along $overline{zy}$ is longer than the path along $overline{xy}$.

Explanation:

Step1: Recall triangle - side length relationship

In a triangle, the side opposite the larger angle is longer.

Step2: Analyze each triangle

In the left - hand triangle with sides $\overline{AB}$, in the middle triangle with sides $\overline{ZY}$, and in the bottom - hand triangle with sides $\overline{XY}$.
For the triangle with side $\overline{AB}$, the angle opposite it has measure $30^{\circ}$. For the triangle with side $\overline{ZY}$, the angle opposite it has measure $65^{\circ}$. For the triangle with side $\overline{XY}$, the angle opposite it has measure $75^{\circ}$.
Since $30^{\circ}<65^{\circ}<75^{\circ}$, the side opposite the $30^{\circ}$ angle ($\overline{AB}$) is the shortest, the side opposite the $65^{\circ}$ angle ($\overline{ZY}$) is of medium length, and the side opposite the $75^{\circ}$ angle ($\overline{XY}$) is the longest.

Answer:

The path along $\overline{XY}$ is longer than the path along $\overline{ZY}$, and the path along $\overline{ZY}$ is longer than the path along $\overline{AB}$.