D is point on BC of ∆ ABC where BD = (1/2)DC. Find the ratio of areas of ∆ ABD :  ∆ ABC.

PQ is a chord in the minor segment of a circle and R is a point on the minor arc PQ. The tangents at the points P and Q meet at the point T. If ∠PRQ =102°, then the measure of ∠PTQ is :

A circle is inscribed in a right triangle with legs of length 5 and 12. What is the radius of the circle?

The length of each of two tangents drawn from an external point to a circle is 10 cm. What is the approximate distance from the external point to the center of the circle, if the radius is 6 cm?

Two circles with radii 6 cm and 2 cm have their centers 10 cm apart. How many common tangents can be drawn between them, and what type(s) of tangents are they?

Two chords AB and CD intersect at a point P inside a circle. If AP = 6 cm, PB = 8 cm, and CP = 4 cm, what is the length of PD?

In a circle, a chord AB is 12 cm long and is at a distance of 4 cm from the center. What is the radius of the circle?

A circle with radius x touches another circle with radius 2x externally. What is the length of a direct common tangent?

Two circles with radii r₁ and r₂ touch each other externally. If the length of their direct common tangent is T, which of the following is the correct relationship between T, r₁ and r₂ ?​

Two circles intersect at two points. Which of the following statements is true?

A chord of a circle has a length of 12 cm. The angle subtended by the chord at a point on the circumference is 30°. What is the distance from the center of the circle to the chord?