Question

question 4 (multiple choice worth 1 points) b2_63r_mc jungle gym monkey bars beth is planning a playground and has decided to place the swings in such a way that they are the same distance from the jungle gym and the monkey bars. if beth places the swings at point d, how could she prove that point d is equidistant from the jungle gym and monkey bars? if $overline{ac}congoverline{bc}$, then point d is equidistant from points a and b because congruent parts of congruent triangles are congruent if $overline{ad}congoverline{cd}$, then point d is equidistant from points a and b because a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects. if $overline{ac}congoverline{bc}$, then point d is equidistant from points a and b because a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects. if $overline{ad}congoverline{cd}$, then point d is equidistant from points a and b because congruent parts of congruent triangles are congruent

Explanation:

Step1: Recall perpendicular - bisector property

A point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment. In the figure, if $\overline{AC}\cong\overline{BC}$ and $DC\perp AB$, then $DC$ is the perpendicular bisector of $\overline{AB}$.

Step2: Analyze each option

• Option 1: If $\overline{AC}\cong\overline{BC}$, then point $D$ is equidistant from points $A$ and $B$ because a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects. This is the correct property application.
• Option 2: $\overline{AD}\cong\overline{CD}$ does not imply that $D$ is equidistant from $A$ and $B$. The congruence of $\overline{AD}$ and $\overline{CD}$ has no relation to the equidistance of $D$ from $A$ and $B$.
• Option 3: The reasoning in this option is correct but the statement $\overline{AC}\cong\overline{BC}$ is mis - associated with the congruent - parts - of - congruent - triangles property instead of the perpendicular - bisector property.
• Option 4: $\overline{AD}\cong\overline{CD}$ does not prove that $D$ is equidistant from $A$ and $B$ based on the congruent - parts - of - congruent - triangles property.

Answer:

If $\overline{AC}\cong\overline{BC}$, then point $D$ is equidistant from points $A$ and $B$ because a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects.