Question

In quadrilateral ABCD, points M, N, P, Q, R, S are the midpoints of sides AB , BC , CD , DA , AC , and BD . Prove that △MQR≅△PNS

Answer 1

Answer:

Applying mid point theorm, △MQR≅△PNS

Step-by-step explanation

As there are all midpoints in the problem, we will try using midpoint theorm.

Mid point theorm states that, when the two midpoints of a triangle are joined, the corresponding side is parallel to third side and is half the length of third side.

As given in the figure, join the points to construct triangle.

Let side MR be x, MQ be y and QR be z.

In triangle ABC, applying mid point theorm, BC = $2x$.

Again in triangle BCD applying mid point theorm, PS = $x$.

In triangle ADC, applying mid point theorm, DC = $2z$.

Again in triangle BDC applying mid point theorm, NS = $z$.

In triangle ABD, applying mid point theorm, BD = $2y$.

Again in triangle BCD applying mid point theorm, NP = $y$.

Thus, corresponding 3 sides are equal and, △MQR≅△PNS