Question

question 1 - 7
mare wants to draw a line perpendicular to $overline{ab}$ passing through point $p$. the figure below shows her work.
she made the following claims about her work:
claim i: $overline{pa}$ is congruent to $overline{pb}$.
claim ii: $overline{aq}$ is congruent to $overline{bq}$.
which statement about the claims is correct?
only claim i is correct

Explanation:

Step1: Recall perpendicular - bisector properties

When a line is perpendicular to a line segment $\overline{AB}$ and passes through its mid - point, any point on the perpendicular bisector is equidistant from the endpoints of the line segment. Here, the line through $P$ is perpendicular to $\overline{AB}$.

Step2: Analyze Claim I

Since the line through $P$ is perpendicular to $\overline{AB}$, and assuming the construction is a perpendicular bisector (or using the property of right - angled congruent triangles), in right - triangles $\triangle PAO$ and $\triangle PBO$ (where $O$ is the intersection of the perpendicular from $P$ to $\overline{AB}$), by the Hypotenuse - Leg (HL) congruence criterion (if the perpendicular bisects $\overline{AB}$) or other congruence rules, $PA = PB$, so $\overline{PA}\cong\overline{PB}$.

Step3: Analyze Claim II

Since the line through $P$ is perpendicular to $\overline{AB}$, and assuming the construction is a perpendicular bisector, the perpendicular bisector of a line segment $\overline{AB}$ divides $\overline{AB}$ into two equal parts. So, $AQ = BQ$, and $\overline{AQ}\cong\overline{BQ}$.

Answer:

Both Claim I and Claim II are correct.