Question

a student is asked to construct $overline{pq}$ which is congruent to $overline{ab}$. he first draws a line and then plots point $p$ on it. using point $p$ as the center, he draws an arc on the line. what is the radius of the arc drawn from point $p$? 1.75 inches 3.5 inches 7 inches 10.5 inches

Explanation:

Step1: Understand the construction goal

The goal is to construct $\overline{PQ}$ congruent to $\overline{AB}$. When constructing a congruent line - segment, we use the length of the given line - segment as the radius of the arc. Since no length of $\overline{AB}$ is given in the question, we assume that we are using the length of $\overline{AB}$ as the radius. But among the options, we need to think about the logic of congruent - segment construction. To construct a line - segment $\overline{PQ}$ congruent to $\overline{AB}$, we set the compass width (radius of the arc) equal to the length of $\overline{AB}$. Without other information, we assume the radius of the arc drawn from point $P$ is the length of $\overline{AB}$. If we consider the general construction process, we take the length of the given segment $\overline{AB}$ as the radius. Since no specific length of $\overline{AB}$ is provided in the problem - context, we assume the radius of the arc is the length of the segment we want to replicate.

Step2: Analyze the options

Since we want to make $\overline{PQ}\cong\overline{AB}$, the radius of the arc drawn from point $P$ should be the length of $\overline{AB}$. But if we assume some reasonable values for construction, and without any other data about $\overline{AB}$, we know that to construct a congruent segment, the radius of the arc is the length of the segment we are copying.

Answer:

There is not enough information in the problem to determine the exact value from the given options. If we assume the length of $\overline{AB}$ is among the options, we need the length of $\overline{AB}$ to make a proper selection. If we have to choose randomly without any additional context, we cannot be sure of the correct answer. But conceptually, the radius of the arc should be equal to the length of $\overline{AB}$.