Question

In the given figure, equal chords AB and CD of a circle with centre O cut each other at right angles at E. If M and N are the mid-points of AB and CD respectively, then OMEN is a

• A. Rectangle
• B. Square
• C. Rhombus
• D. none of these

Answer 1

The correct option is

B

Square

In ΔOME and ΔONE,

OM =ON [equal chords are equidistant from the centre]

OE = OE [Common side]

∠OME = ∠ONE = 90°

$\therefore$ ∠OME ≅ ∠ONE [by RHS congruency]

⇒ ME = NE [by CPCT]

Conside quadrilateral OMEN, ∠MEN = 90° [ as the chords AB and CD cut each other at right angles]

∠MON = 360° - (∠OME + ∠MEN + ∠ONE)

= 360° - (90° + 90° + 90°) = 90° [∠MEN = 90°, given]

Thus, in quadrilateral OMEN,

OM =ON , ME = NE

and ∠OME = ∠ONE = ∠MEN = ∠MON = 90°

$\therefore$, OMEN is a square.