Mathematics
In the given circle with centre O, angle ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.
Answer
From figure,
⇒ ∠ADC + ∠ABC = 180° [Sum of opposite angles in a cyclic quadrilateral = 180°]
⇒ ∠ADC + 100° = 180°
⇒ ∠ADC = 180° - 100°
⇒ ∠ADC = 80°.
In △ADC,
⇒ ∠ADC + ∠CAD + ∠ACD = 180° [By angle sum property of triangle]
⇒ 80° + ∠CAD + 40° = 180°
⇒ ∠CAD = 180° - 120° = 60°.
From figure,
⇒ ∠DCT = ∠CAD = 60° [Angles in alternate segment are equal].
Hence, ∠DCT = 60° and ∠ADC = 80°.
Related Questions
In the figure, given below, AC is a transverse common tangent to two circles with centers P and Q and of radii 6 cm and 3 cm respectively. Given that AB = 8 cm, calculate PQ.
In the figure, given below, O is the center of the circumcircle of triangle XYZ. Tangents at X and Y intersect at point T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.
In the figure given below, O is the center of the circle and SP is a tangent. If ∠SRT = 65°, find the values of x, y and z.
(i) A tangent to a circle is a line coplanar with the circle which meets the circle at exactly one point.
(ii) When two circles touch each other at exactly one point, either externally or internally, they are said to be tangent to each other.
AB is the common tangent to both the circles that are tangent to each other.
Based on the above information, how many common tangents do you think can be drawn for two circles, when the circles are as given below ? Draw the tangent(s) in each case.
