Geometry

In the figure, PQ and PR are tangents to the circle with center O. If angle QPR = 40 degrees, find angle QOR.

If the length of a chord of a circle of radius 10 cm is 16 cm, find its distance from the center.

In the figure, O is the center and AB is a diameter. C is a point on the circle such that angle ABC = 30 degrees. Find angle BAC.

If the angle between two tangents drawn from an external point P to a circle is 120 degrees and the radius is 10 cm, find the distance of the point P from the center.

A circle is inscribed in a triangle ABC with sides 8 cm, 10 cm, and 12 cm. Find the length of the segment of the side 12 cm from the vertex opposite to the side 10 cm.

Find the area of the segment of a circle of radius 14 cm if the central angle is 90 degrees.

Two circles touch each other externally at point T. A direct common tangent AB touches the circles at A and B. Find the measure of angle ATB.

A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point on the major arc.

In a circle of radius 5 cm, AB and CD are two parallel chords of length 8 cm and 6 cm respectively. If they lie on opposite sides of the center, find the distance between them.

In the figure, O is the center. AB is a chord and PT is the tangent at T. If PT = 12 cm and the distance of P from the center is 13 cm, find the radius.

If a cyclic trapezium has its non-parallel sides equal, what kind of trapezium is it?

In a circle, two chords AB and CD intersect at a point E outside the circle. If AB = 5, BE = 4, and DE = 6, find the length of CD.

The length of the common chord of two intersecting circles is 24 cm. If their radii are 13 cm and 15 cm, find the distance between their centers.

The area of a sector is 1/12 of the area of the circle. What is the central angle of the sector?

An arc of length 20 cm subtends an angle of 2 radians at the center. What is the radius of the circle?

If the angle between two tangents from P to a circle is 90 degrees and the radius is 5 cm, what is the length of each tangent?

In a cyclic quadrilateral ABCD, side AB is the diameter. If angle ADC = 130 degrees, find angle BAC.

Find the area of the largest triangle that can be inscribed in a semi-circle of radius 'r'.

If a circle passes through the vertices of a triangle with sides 3, 4, and 5, what is the radius of the circle?

Two circles touch internally. The sum of their areas is 170*pi and the distance between their centers is 4. Find the radius of the larger circle.