Question

segment lp is in △lmn.
. $overline{lp}perpoverline{nm}$
. the length of $np = 2.4$ cm.
. the length of $pm = 2.4$ cm.
which statement is true?
a. $mangle nlpcong mangle lnp$
b. $overline{lp}congoverline{mn}$
c. $mangle plmcong mangle lmp$
d. $overline{ln}congoverline{lm}$

Explanation:

Step1: Recall right - triangle and congruence concepts

We know that $\overline{LP}\perp\overline{NM}$, $NP = PM=2.4$ cm. In right - triangles $\triangle LNP$ and $\triangle LMP$, $LP$ is common, $NP = PM$, and $\angle LPN=\angle LPM = 90^{\circ}$.

Step2: Apply the Side - Angle - Side (SAS) congruence criterion

By the SAS congruence criterion, $\triangle LNP\cong\triangle LMP$.

Step3: Use corresponding parts of congruent triangles

Since $\triangle LNP\cong\triangle LMP$, their corresponding sides and angles are congruent. So, $\overline{LN}\cong\overline{LM}$ (corresponding sides of congruent triangles).

Answer:

D. $\overline{LN}\cong\overline{LM}$