Question

Assertion: Two diameters of a circle intersect each other at right angles. Then the quadrilateral formed by joining their end-points is a square.

Reason: Angle in a semi-circle is a right angle.

Which of the following options is correct?

• A. Both assertion and reason are true and reason is the correct explanation of assertion.
• B. Both assertion and reason are true, but reason is not the correct explanation of assertion.
• C. Assertion is true, but the reason is false.
• D. Assertion is false, but the reason is true.

Answer 1

The correct option is A

Both assertion and reason are true and reason is the correct explanation of assertion.

Let AB and CD be two perpendicular diameters of a circle with centre O.
In $\Delta s$ AOC and BOC, we have
OA = OB. [O is the midpoint of diameter AB]
$\angle AOC=\angle BOC$ $(={90}^{\circ })$ [AB is perpendicular to CD]
And, OC = OC [common side]
$⟹\Delta AOC\cong \Delta BOC$ [By S.A.S. congruence]
By CPCT, $AC=BC.$
Also, $\angle ACB={90}^{\circ }$ [Angle in a semi-circle is a right angle]
Similarly, we get
$BC=BD$ and $\angle CBD={90}^{\circ },$ and
$DB=DA$ and $\angle BDA={90}^{\circ }.$
Thus, ACBD has all its sides and angles equal, and hence is a square.