Question

Show that the quadrilateral formed by joining the midpoints of the pairs of adjecent sides of a rhombus is a rectangle.

Answer 1

Step-by-step explanation:

consider ∆ ABC

we know that p and q are the midpoints of AB and BC

by using midpoint theorem

we know that PQ || AC and PQ = 1/2AC

consider ∆ ADC

we know that RS || AC and RS = 1/2AC

it can be written as PQ|| RS and RS = RS= 1/2AC ..eq ..1

consider ∆ BAD

we know that P and S are the midpoints of AB and AD

based on the mid point theorem

we know that PS || BD and PS = 1/2DB

consider ∆ BCD

we know that RQ || BD and RQ = 1/2DB

it can be written as PS = RQ = 1/2DB .. eq...2

by eq 1 and 2

the diagonals intersects at right angles in a rohmbus so we get

∠EQF = 90°

we know that RQ || DB

so we get RE || FO

in the same way SR || AC

so we get FR || OE

so we know that OERF is a parllelogram

we know that the opposite angles are equal in a parllelogram ,

so we get

∠FRE = ∠EOF = 90°

so we know that PQRS is a parllelogram

having ∠R = 90°

therefore it is proved that the quadrilateral formed by joining the midpoints of the pairs of adjecent sides of a rhombus is a rectangle.