Concept of Circle

In the following figure, OABC is a square. A circle is drawn with O as centre which meets OC at P and OA at Q.
Prove that:
( i ) ΔOPA ≅ ΔOQC 
( ii ) ΔBPC ≅ ΔBQA

Draw a circle of radius 6 cm. In the circle, draw a chord AB = 6 cm.

(i) If O is the center of the circle, join OA and OB.

(ii) Assign a special name to ∆AOB

(iii) Write the measure of angle AOB.

Two circles touch internally. The sum of their areas is 116 π cm² and the distance between their centres is 6 cm. Find the radii of the circles ?

In fig. there are two concentric circles with Centre O of radii 5cm and 3cm. From an
external point P, tangents PA and PB are drawn to these circles if AP = 12cm, find the
tangent length of BP.

State, if the following statement is true or false:

Every diameter bisects a circle and each part of the circle so obtained is a semi-circle.

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.

Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that the radius of the circle is 7 cm.

In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm such that the segments BC and DC into which BC is divided by the point of contact D, are of
lengths 6cm and 9cm respectively. If the area of 2 ΔABC = 54cm² then find the lengths of sides AB and AC.