Final answer:
By using the given information that N is the midpoint of MP, QN is orthogonal to SM, RN is orthogonal to SP, and QN is congruent to RN, we deduce that triangles QNP and RNM are congruent. This leads to the conclusion that ∆MSP is isosceles, as the corresponding sides SM and SP must be equal in length.
Step-by-step explanation:
To prove that ∆MSP is isosceles given N is the midpoint of MP, QN⊥SM, RN⊥SP, and QN≅RN, we can start by noticing that triangles QNP and RNM are right triangles. Since N is the midpoint of MP and QN is congruent to RN, we have two congruent sides (QN and RN) and a shared side (NP in ∆QNP and NM in ∆RNM). By the SAS (Side-Angle-Side) Congruence Theorem, the two right triangles are congruent. Because these triangles are congruent, their corresponding parts are also congruent. Therefore, QM is congruent to RM, which are the hypotenuses of triangles QNP and RNM, respectively.
This implies that the sides SM and SP of triangle ∆MSP are equal in length, as they are the sum of congruent segments (with NM shared by both triangles), making ∆MSP an isosceles triangle.