Question

proof a city planner is designing a new park. the park has two straight paths, \\(\overline{ab}\\) and \\(\overline{cd}\\), which are the same length. a monument, m, is located at the midpoint of both paths.
a. the city planner thinks that the length of \\(\overline{am}\\) will be the same as the length of \\(\overline{cm}\\). explain why this makes sense.
both segments are half the length of \\(\boxed{\text{select choice}}\\), so the lengths of the shorter segments must be \\(\boxed{\text{select choice}}\\).

Explanation:

Step1: Recall Midpoint Definition

A midpoint of a segment divides it into two equal - length parts. For segment \(AB\), since \(M\) is the midpoint, \(AM = MB\) and \(AM=\frac{1}{2}AB\). Similarly, for segment \(CD\), since \(M\) is the midpoint, \(CM = MD\) and \(CM=\frac{1}{2}CD\).

Step2: Use Given Information

We are given that \(AB = CD\) (the two paths are the same length). So, \(\frac{1}{2}AB=\frac{1}{2}CD\). Substituting the expressions from Step 1, we get \(AM = CM\) because \(AM=\frac{1}{2}AB\) and \(CM=\frac{1}{2}CD\) and \(AB = CD\). Also, the shorter segments ( \(AM\) and \(CM\)) must be equal because they are both half of equal - length segments (\(AB\) and \(CD\) respectively).

Answer:

The city planner is correct because \(M\) is the midpoint of both \(\overline{AB}\) and \(\overline{CD}\). By the definition of a midpoint, \(AM=\frac{1}{2}AB\) and \(CM = \frac{1}{2}CD\). Since \(AB = CD\), then \(\frac{1}{2}AB=\frac{1}{2}CD\), so \(AM = CM\). The shorter segments (\(AM\) and \(CM\)) must be equal as they are half of equal - length segments (\(AB\) and \(CD\)).