Question : In the given figure, AC and DE are perpendicular to the tangent CB. AB passes through the centre O of the circle, whose radius is 20 cm. If AC is 36 cm, what is the length (in cm) of DE?
Option 1: 4
Option 2: 6
Option 3: 2
Option 4: 8
Correct Answer: 4
Solution : Given: AC $\perp$ CB DE $\perp$ CB OP $\perp$ CB ( radius $\perp$ tangent) AC || DE || OP AO = OD = OP= radius = 20 cm AC = 36 cm Let BD = $x$ cm AB = $x + 20 + 20$ = $x$+40 OB = $x + 20$ Since $\angle$ A = $\angle$ O = $\angle$ D and $\angle$ P = $\angle$ C = $\angle$ E, $\triangle$ OBP ~ $\triangle $ABC ~ $\triangle $DBE $\frac{OB}{OP}$ = $\frac{AB}{AC}$ ⇒ $\frac{x+20}{20}$ = $\frac{x+40}{36}$ ⇒ $36x+720$ = $20x+800$ ⇒ $16x$ = 80 ⇒ BD = $x$ = 5 ⇒ $\frac{AB}{AC}$ = $\frac{DB}{DE}$ ⇒ $\frac{5+40}{36}$ = $\frac{5}{DE}$ ⇒ DE = $\frac{36}{45}×5$ = 4 cm Hence, the correct answer is 4.
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Question : AB is a chord in a circle of radius 13 cm. From centre O, a perpendicular is drawn through AB intersecting AB at point C. The length of OC is 5 cm. What is the length of AB?
Option 1: 24 cm
Option 2: 12 cm
Option 3: 20 cm
Option 4: 15 cm
Question : In the given figure, point O is the centre of a circle of radius 13 cm and AB is a chord perpendicular to OD. If CD = 8 cm, what is the length (in cm) of AB?
Option 1: 6 cm
Option 3: 24 cm
Option 4: 28 cm
Question : A sector of a circle has a central angle of 45° and an arc length of 22 cm. Find the radius of the circle. ( Use $\pi=\frac{22}{7}$)
Option 1: 32 cm
Option 2: 35 cm
Option 3: 28 cm
Option 4: 36 cm
Question : In a given circle, the chord PQ is of length 18 cm. AB is the perpendicular bisector of PQ at M. If MB = 3 cm, then the length of AB is:
Option 1: 27 cm
Option 2: 30 cm
Option 4: 25 cm
Question : Out of two concentric circles, the radius of the outer circle is 6 cm and the chord PQ of the length 10 cm is a tangent to the inner circle. Find the radius (in cm) of the inner circle.
Option 1: $4$
Option 2: $\sqrt{7}$
Option 3: $\sqrt{13}$
Option 4: $\sqrt{11}$
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