Mathematics
In the figure (i) given below, P and Q are centers of two circles intersecting at B and C. ACD is a straight line. Calculate the value of x.
Answer
Arc AB subtends ∠APB at center and ∠ACB at the point C on the circle.
∴ ∠APB = 2∠ACB
⇒ ∠APB = 2∠ACB
⇒ 130° = 2∠ACB
⇒ ∠ACB = 65°.
From figure,
∠ACB + ∠BCD = 180°. (∵ both angles form a linear pair)
⇒ 65° + ∠BCD = 180°
⇒ ∠BCD = 180° - 65°
⇒ ∠BCD = 115°.
In circle with center Q,
⇒ ∠BQD + Reflex ∠BQD = 360°
⇒ x° + Reflex ∠BQD = 360°
⇒ Reflex ∠BQD = 360° - x°.
Arc BD subtends reflex ∠BQD at center and ∠BCD at the point C on the circle.
∴ Reflex ∠BQD = 2∠BCD
⇒ 360° - x° = 2 × 115°
⇒ 360° - x° = 230°
⇒ x° = 360° - 230°
⇒ x° = 130°.
Hence, the value of x = 130.
Related Questions
In the figure (i) given below, chord ED is parallel to the diameter AC of the circle. Given ∠CBE = 65°, calculate ∠DEC.
In the figure (ii) given below, O is the circumcenter of triangle ABC in which AC = BC. Given that ∠ACB = 56°, calculate
(i) ∠CAB
(ii) ∠OAC.
In the figure (ii) given below, it is given that ∠ABC = 40° and AD is a diameter of the circle. Calculate ∠DAC.
In the figure (i) given below, AB is a diameter of the circle APBR. APQ and RBQ are straight lines, ∠A = 35°, ∠Q = 25°. Find :
(i) ∠PRB
(ii) ∠PBR
(iii) ∠BPR
