Question

Assertion: The circle drawn taking any one of the equal sides of an isosceles right triangle as diameter bisects the base.
Reason: The angle in a semicircle is 1 right angle.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

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

Assertion (A): Let ABC be a triangle in which AB = AC and let O be the mid point of AB. With O as centre and OA as radius, draw a circle meeting BC at D.

In Δ ADB, ∠ADB =90° (Angle in a semicircle)
Also, ∠ADB + ∠ADC = 180°
⇒ ∠ADC = 90°
In ΔADB and ΔADC, we have:
AB = AC (Given)
AD = AD (Common)
and ∠ADB = ∠ADC = 90°
∴ ΔADB ≅ ΔADC (RHS criterion)
∴ BD = DC
Hence, the given circle bisects the base.
Thus, assertion (A) is true.
Reason (R): Let ∠BAC be an angle in a semicircle with centre O and diameter BC.

Now, the angle subtended by arc BOC at the centre is ∠BOC = 180°
∴ ∠BOC = 2∠BAC
⇒ 2∠BAC = 180°
⇒ ∠BAC = 90°
Thus, reason (R) is true.
Clearly, reason (R) gives assertion (A).
Hence, the correct option is (a).