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Circles

Multiple Choice Questions

311. A line intersecting a circle in two points is called a ___________.

Diameter
Chord
Secant
Secant

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Short Answer Type

312. In the given Fig. 10.98, a circle is inscribed in a triangle A B C having side BC = 8 cm, AC = 10 cm and AB = 12 cm. Find AD, BE and CF.

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313. ΔABC is inscribed in a circle. Find the length BC.

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314. A point A is 13 cm from the centre of the circle. The length of the tangent drawn from A to the circle is 12 cm. Find the radius, of the circle.

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315. In figure, if ∠ATO = 40°, find ∠AOB.

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316.

In Fig. , PQ is tangent at point C to a circle with centre O. If AB is a diameter and ∠CAB = 30°, find ∠PCA.

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317.

In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

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318.
In Fig, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP=2r, show that ∠ OTS = ∠ OST = 30°.

In the given figure,

Given
OP= 2r
∠OTP = 90° (radius drawn at the point of contact is perpendicular to the tangent)
Now ,
In ΔOTP,
Sin∠ OPT = OT/OP = 1/2 = Sin 30^o
⇒ ∠ OPT = 30^o
therefore,
∠ TOP=60^o
∴ ΔOTP is a 30^o-60^o-90^o, right triangle.

In ΔOTS,
OT = OS … (Radii of the same circle)
therefore,
ΔOTS is an isosceles triangle.

∴∠OTS = ∠OST … (Angles opposite to equal sides of an isosceles triangle are equal)

In ΔOTQ and ΔOSQ

OS = OT … (Radii of the same circle)
OQ = OQ ...(side common to both triangles)
∠OTQ = ∠OSQ … (angles opposite to equal sides of an isosceles triangle are
equal)

∴ ΔOTQ ≅ ΔOSQ … (By S.A.S)
∴ ∠TOQ = ∠SOQ = 60° … (C.A.C.T)
∴ ∠TOS = 120° … (∠TOS = ∠TOQ + ∠SOQ = 60° + 60° = 120°)
∴ ∠OTS + ∠OST = 180° – 120° = 60°
∴ ∠OTS = ∠OST = 60° ÷ 2 = 30°

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