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
7 cm
12 cm
15 cm
24.5 cm
In Fig. 10.11, if TP and TQ are two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to
60°
70°
80°
90°
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, then ∠POA is equal to
50°
60°
70°
80°
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
The tangents at the endpoints of a diameter are perpendicular to the radii at those points. Since the radii lie along a straight line, the tangents are parallel.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
The radius drawn to the point of contact is perpendicular to the tangent at that point. Hence this perpendicular must pass through the centre.
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Using AO² = AP² + r² → 5² = 4² + r² → r = 3 cm.
Two 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.
Chord length = 2√(5² − 3²) = 2√16 = 8 cm.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
From tangents drawn from an external point, tangents to points of contact are equal. Adding pairs of equal tangents gives AB + CD = AD + BC.
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle and AB is another tangent intersecting XY at A and X′Y′ at B. Prove ∠AOB = 90°.
Opposite tangents form supplementary angles; geometry of radii and symmetry gives ∠AOB = 90°.
Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
If ∠AOB is at centre, then angle between tangents = 180° − ∠AOB.
Prove that the parallelogram circumscribing a circle is a rhombus.
Opposite sides of a circumscribed quadrilateral sum equally; hence consecutive sides of parallelogram are equal → rhombus.
A triangle ABC circumscribes a circle of radius 4 cm with BD = 8 cm and DC = 6 cm. Find AB and AC.
Using tangent properties: AB = s − c = 12 cm, AC = s − b = 10 cm.
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre.
Tangent lengths imply opposite arcs sum to 180°, hence the angles subtended at the centre are supplementary.