Question

constructing parallel and perpendicular lines quick check select the choice that indicates $overleftrightarrow{ab}$ is the perpendicular bisector of $overleftrightarrow{cd}$ if they intersect at $p$. (1 point) $\bigcirc overline{ap} cong overline{bp}$ $\bigcirc overline{ac} perp overline{bd}$ and $overline{ap} cong overline{bp}$ $\bigcirc overline{cd} perp overline{ab}$ $\bigcirc overline{ab} perp overline{cd}$ and $overline{cp} cong overline{dp}$

Explanation:

A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two congruent parts. So, for \(\overleftrightarrow{AB}\) to be the perpendicular bisector of \(\overleftrightarrow{CD}\) (intersecting at \(P\)), we need \(\overline{AB}\perp\overline{CD}\) (perpendicular) and \(\overline{CP}\cong\overline{DP}\) (bisects \(CD\)).

• The first option only shows \(AP\cong BP\), which is about bisecting \(AB\), not \(CD\).
• The second option involves \(AC\) and \(BD\), which are not relevant.
• The third option only shows perpendicularity, not bisecting \(CD\).
• The fourth option has both perpendicularity (\(\overline{AB}\perp\overline{CD}\)) and bisecting (\(\overline{CP}\cong\overline{DP}\)).

Answer:

\(\boldsymbol{\overline{AB} \perp \overline{CD}}\) and \(\boldsymbol{\overline{CP} \cong \overline{DP}}\) (the last option)