A rectangle is formed by joining the mid-points of the sides of a rhombus. Show that the area of rectangle is half the area of rhombus.

Let the diagonal of the rhombus be P and Q. Then Area of rhombus = 1/2 (P*Q) The Rectangle formed by joining the mid-points of the sides of rhombus will have dimensions of P/2 and Q/2 We can prove this by using mid-point thereon of triangle .(The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side.) So the side of the rectangle is P/2 and Q/2 Area of rectangle = PQ/4 Which is half the area of the rhombus. Hence proved. read less

Given :- ABCD is a rectangle and P,Q,R,S are their midpoints. To Prove:- PQRS is a rhombus. Proof:- In Tri. ABC, P and Q are the mid points . So, PQ is parallel AC and PQ = 1/2 *AC (the line segment joining the mid points of 2 sides of the triangle is parallel to the third side and half of the third side) Similarly RS is parallel AC and RS = 1/2* AC Hence, both PQ and RS are parallel to AC and equal to 1/2*AC. Hence, PQRS is a parallelogram In triangles APS & BPQ, AP=BP (P is the mid point of side AB) angle PAS = angle PBQ (90 degree each) and, AS = BQ (S and Q are the mid points of AD and BC respectively and since oppsite sides of a rectangle are equal, so their halves will also be equal) triangle APS is congruent to triangle BPQ (By SAS) So, PS=PQ (By CPCT) PQRS is a parallelogram in which adjacent sides are equal, ...PQRS is a rhombus read less