Advertisements
Advertisements
Question
In the following figure, O is the centre of the circle, BD = OD and CD ⊥ AB. Find ∠CAB.
Advertisements
Solution
Given, in the figure BD = OD, CD ⊥ AB.
In ΔOBD, BD = OD ...[Given]
OD = OB ...[Both are the radius of circle]
∴ OB = OD = BD
Thus, ΔODB is an equilateral triangle.
∴ ∠BOD = ∠OBD = ∠ODB = 60°
In ΔMBC and ΔMBD,
MB = MB ...[Common side]
∠CMB = ∠BMD = 90°
And CM = MD ...[In a circle, any perpendicular drawn on a chord also bisects the chord]
∴ ΔMBC ≅ ΔMBD ...[By SAS congruence rule]
∴ ∠MBC = ∠MBD ...[By CPCT]
⇒ ∠MBC = ∠OBD = 60° ...[∵ ∠OBD = 60°]
Since, AB is a diameter of the circle.
∴ ∠ACB = 90°
In ΔACB, ∠CAB + ∠CBA + ∠ACB = 180° ...[By angle sum property of a triangle]
⇒ ∠CAB + 60° + 90° = 180°
⇒ ∠CAB = 180° – (60° + 90°) = 30°
APPEARS IN
RELATED QUESTIONS
If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC ?
In the given figure, if ABC is an equilateral triangle. Find ∠BDC and ∠BEC.
In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If \[\angle\] AOQ = 58º, find \[\angle\] ATQ.
Use the figure given below to fill in the blank:
Diameter of a circle is ______.
The center of a circle is at point O and its radius is 8 cm. State the position of a point P (point P may lie inside the circle, on the circumference of the circle, or outside the circle), when:
(a) OP = 10.6 cm
(b) OP = 6.8 cm
(c) OP = 8 cm
Find the radius of the circle
Diameter = 76 cm
A point P is 10 cm from the center of a circle. The length of the tangent drawn from P to the circle is 8 cm. The radius of the circle is equal to ______
In the following figure, O is the centre of the circle. Name a chord, which is not the diameter of the circle.
What is the name for the perimeter of a circle?
