Fill In the Blanks
Short Answer TypeA tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(a) 12 cm (b) 13 cm
(c) 8.5 cm (d) 
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm
(C) 15 cm (D) 24.5 cm.
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to
(A) 60° (B) 70°
(C) 80° (D) 90°
Fig. 10.11
Long Answer TypeIf tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to
(A) 50° (B) 60°
(C) 70° (D) 80°.
Short Answer TypeProve that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Long Answer TypeTwo concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Let O be the common centre of the two concentric circle.
Let PQ be a chord of the larger circle which touches the smaller circle at M.
Join OM and OP.
Since, the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Therefore,
∠OMP = 90°
Now,
In ΔOMP, we have
OP2 = OM2 + PM2
[Using Pythagoras theorem]
⇒ (5)2 = (3)2 + PM2
⇒ 25 = 9 + PM2
⇒ PM2 = 16
⇒ PM = 4 cm
Since, the perpendicular from the centre of a circle to a chord bisects the chord.
Therefore,
PM = MQ = 4 cm
∴ PQ = 2 PM = 2 x 4 = 8 cm
Hence, the required length = 8 cm.
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