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 : Â
(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.
Since, the tangent at any point of a circle is perpendicular to radius through the point of contact.
Therefore, ∠OPQ = 90°
It is given that OQ = 5 cm
and    PQ = 4 cm
In right ΔOPQ, we have
OQ2Â = OP2Â + PQ2
[Using Pythagoras Theorem]
OP2 = OQ2 – PQ2
⇒ OP2 = (5)2 – (4)2
= 25 – 16 = 9
⇒ OP = 3 cm
Hence, the radius of the circle is 3 cm.
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.
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