In the figure, ㄥACP=40 and ㄥBPD=120, then ㄥCBD=____________

Figure,O is centre of the circle and PA=PB Find ㄥOPA

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

ABCD is a cyclic quadrilateral whose diagonals interest at a point E. If \(\angle DBC=70°, \angle BAC\) is 40°, find \(\angle BCD\) . Further, if AB=BC, find \(\angle ECD\) .

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Two chords PQ and RS of a circle are parallel to each other and AB is the perpendicular bisector of PQ. Without using any construction, prove that AB bisects RS.

In the figure, diameter AB and a chord AC have a 'common end point A. If the length of AB is 20 cm and of AC is 12 cm, how far is AC from the centre of the circle?

Bisector AD of \(\angle BAC\) of \(\Delta ABC\) passes through the centre of the circumcircle of \(\Delta ABC\) as shown in the figure. Prove that AB = AC.

D and E are respectively the points on equal sides AB and AC of an isosceles triangle ABC such that B, C, E and D are concylic, as shown in the given figure, if 0 is the point of intersection of CD and BE, prove that AO is the bisector of the line segment DE.

Raja, Renu and Reena are three friends. They decided to sweep a circular park near their homes. They divided the park into three parts by two equal chords AB and AC for convenience.
(i) Prove that the centre of the park lies on the angle bisector of \(\angle\)BAC
(ii) Which mathematical concept is used in the above problem?
(iii) By deciding sweeping, which value is depicted by the three friends?

Mr. Mehta, owner of a biscuit manufacturing company, wants to stick butter on a circular biscuit, in the form of two equal chords. He wishes the length of each chord should be more than the radius and less than the diameter of the biscuit. Assuming
that the thickness of the biscuits is negligible.
(i) Prove that the butter-chords subtend equal angles at the centre of the biscuit.
(ii) What is the measure-range of the angle subtended by either butter-chord at the centre?
(iii) Which mathematical concept is used in the above problem?
(iv) Which value is depicted by Mr. Mehta as an owner of a manufacturing company?