Mathematics
Points A, C, B and D are concyclic, AB is diameter and ∠ABC = 60°.
Assertion (A) : ∠BAC = 60°.
Reason (R) : AB is diameter so ∠ACB = 90° and ∠ABC + ∠BAC = 90°.
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.
Answer
Join AC.
Since AB is diameter, that means :
∠ACB = 90° (Angles in a semicircle is a right angle)
In △ ABC, using angle sum property,
⇒ ∠ABC + ∠ACB + ∠BAC = 180° ………………..(1)
⇒ 60° + 90° + ∠BAC = 180°
⇒ 150° + ∠BAC = 180°
⇒ ∠BAC = 180° - 150°
⇒ ∠BAC = 30°
So, assertion (A) is false.
From equation (1),
⇒ ∠ABC + 90° + ∠BAC = 180°
⇒ ∠ABC + ∠BAC = 180° - 90°
⇒ ∠ABC + ∠BAC = 90°
So, reason (R) is true.
Hence, option 2 is the correct option.
Related Questions
In the given circle, ∠BAD = 95°, ∠ABD = 40° and ∠BDC = 45°.
Assertion (A) : To show that AC is a diameter, the angle ADC or angle ABC need to be proved to be 90°.
Reason (R) : In △ADB,
∠ADB = 180° - 95° - 40° = 45°
∴ Angle ADC = 45° + 45° = 90°
(i) A is true, R is false
(ii) A is false, R is true
(iii) Both A and R are true and R is correct reason for A
(iv) Both A and R are true and R is incorrect reason for A
ABCD is a cyclic quadrilateral, BD and AC are its diameters. Also, ∠DBC = 50°.
Assertion (A) : ∠BAC = 40°.
Reason (R) : ∠BAC = ∠BDC = 180° - (50° + 90°) = 40°.
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.
AB is diameter of the circle and ∠ACD = 38°.
Assertion (A) : x = 38°.
Reason (R) : ∠ACB = 90°, x = ∠DCB = 90° - 38° = 52°.
A is true, R is false.
A is false, R is true.
Both A and R are true and R is correct reason for A.
Both A and R are true and R is incorrect reason for A.