Question

Which of the following quadrilateral is formed when the mid point of adjacent sides of rectangle are joined with each other?

• A. Rhombus
• B. Square
• C. rectangle
• D. None of these

Answer 1

The correct option is A Rhombus


Let $ABCD$ is the rectangle and $P$, $Q$, $R$ and $S$ are the midpoints of sides $AB,BC,CD\text{and}DA$ respectively.

we know that rectangle is a quadrilateral and when midpoints of adjacents sides of quadrilateral are joined they form parallelogram.
So, $PQRS$ is parallelogram.

But in rectangle, the diagonals are equal to each other.

Here,
$AC=BD$
and also $PQ=\frac{1}{2}AC\text{(applying midpoint theorem in}△ABC)$
$QR=\frac{1}{2}BD\text{(applying mid point theorem in}△BCD)$
or, $QR=\frac{1}{2}AC(\because BD=AC)$
So, $QR=PQ$

Similiarly, in $△ACD\&△ABD$
$RS=\frac{1}{2}ACPS=\frac{1}{2}BD$
$\therefore RS=SP=PQ=QR$

As, in paralleogram $PQRS$ adjacent sides are equal so it is Rhombus.

Hence, option $\left(a\right)$ correct.