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?
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 Δs AOC and BOC, we have
OA = OB. [O is the midpoint of diameter AB]
∠AOC=∠BOC (=90∘) [AB is perpendicular to CD]
And, OC = OC [common side]
⟹ΔAOC≅ΔBOC [By S.A.S. congruence]
By CPCT, AC=BC.
Also, ∠ACB=90∘ [Angle in a semi-circle is a right angle]
Similarly, we get
BC=BD and ∠CBD=90∘, and
DB=DA and ∠BDA=90∘.
Thus, ACBD has all its sides and angles equal, and hence is a square.