Question

examine this figure. which two pieces of information, if true, would help to prove that △lmp≅△nmp by hl? select two options. point p is the mid - point of mk. line mk is the perpendicular bisector of ln. ml≅mp ml≅mn pk≅pk

Explanation:

Step1: Recall HL - Hypotenuse - Leg theorem

HL theorem states that if the hypotenuse and a leg of a right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. In \(\triangle LMP\) and \(\triangle NMP\), \(\overline{MP}\) is common.

Step2: Analyze the hypotenuse condition

For \(\triangle LMP\) and \(\triangle NMP\) to be congruent by HL, we need the hypotenuses to be congruent. The hypotenuses of right - triangles \(\triangle LMP\) and \(\triangle NMP\) are \(\overline{ML}\) and \(\overline{MN}\) respectively. So, \(\overline{ML}\cong\overline{MN}\) is one required piece of information.

Step3: Analyze the right - angle and leg - related condition

If line \(MK\) is the perpendicular bisector of \(\overline{LN}\), then \(\angle MPL=\angle MPN = 90^{\circ}\) and \(\overline{LP}\cong\overline{NP}\). Also, \(\overline{MP}\) is common. This helps in satisfying the HL criterion as we have right - angles and can use the hypotenuse and leg relationships.

Answer:

B. Line MK is the perpendicular bisector of \(\overline{LN}\), D. \(\overline{ML}\cong\overline{MN}\)