Question

the diagram shows $\triangle jlp$. • point $k$ is the midpoint of $\overline{jl}$. • point $q$ is the midpoint of $\overline{pj}$. • $\overline{jm}$ is a median of $\triangle jlp$. which statements are true? select all the correct answers. $\square$ a. $\overline{tj}$ is congruent to $\overline{tk}$. $\square$ b. $\overline{pm}$ is congruent to $\overline{lm}$. $\square$ c. point $t$ is equidistant from points $j$, $l$, and $p$. $\square$ d. point $t$ is the midpoint of each median of $\triangle jlp$. $\square$ e. when a median is drawn through point $q$, it must intersect $\overline{pk}$ and $\overline{jm}$ at point $t$. $\square$ f. there is only one median that can be drawn through point $q$ and it must pass through point $l$.

Explanation:

• Option A: \( \overline{TJ} \) and \( \overline{TK} \) – No info shows \( T \) is midpoint of \( JK \), so not congruent.
• Option B: \( \overline{JM} \) is a median, so \( M \) is midpoint of \( PL \), thus \( \overline{PM} \cong \overline{LM} \) (true).
• Option C: \( T \) is centroid, not equidistant from all vertices (equidistant would be circumcenter, not centroid) – false.
• Option D: Centroid divides medians in 2:1 ratio, not midpoint – false.
• Option E: Medians intersect at centroid \( T \), so median through \( Q \) (midpoint of \( PJ \)) intersects other medians at \( T \) – true.
• Option F: Median through \( Q \) connects \( Q \) to midpoint of \( JL \) (which is \( K \)), not \( L \) – false.

Answer:

B. \( \overline{PM} \) is congruent to \( \overline{LM} \)
E. When a median is drawn through point \( Q \), it must intersect \( \overline{PK} \) and \( \overline{JM} \) at point \( T \)