Question

Four tangents drawn to a circle are extended from both the sides to form a quadrilateral. Which of these quadrilateral is not possible ?

• A. Trapezium
• B. Square
• C. Rectangle
• D. Rhombus

Hint:

A circle can be inscribed inside a quadrilateral if and only if the sum of opposite sides is equal.
For a rectangle of sides \(a\) and \(b\), \(a+a = 2a\) and \(b+b = 2b\). These are equal only when \(a = b\) (which makes it a square).

Answer 1

The Correct Option is C

Step 1: Think about the incircle's centre instead of side-sums.
If a quadrilateral has an inscribed circle, its centre must be equally distant from all four sides, that distance being the radius $r$.
Step 2: Apply this to a rectangle.
For a non-square rectangle with sides $a$ and $b$ ($a \neq b$), a circle touching the two sides of length $a$ needs radius $\frac{b}{2}$, while touching the two sides of length $b$ needs radius $\frac{a}{2}$.
Step 3: Spot the contradiction.
A single circle cannot have two different radii at once, so we would need $\frac{a}{2} = \frac{b}{2}$, i.e. $a=b$, which only happens when the rectangle is actually a square.
Squares and rhombuses trivially satisfy this (all sides equal), and a suitable trapezium can too, but a general rectangle cannot.
\[ \boxed{\text{Rectangle}} \]