The length of the direct common tangent of two circles with radius $12 cm$ and $4 cm$ is $15 cm$. Find the distance between the centres of the circles.

The distance between the centres of two circles with radius $8 cm$ and $3 cm$ is $13 cm$. What is the length of the direct common tangent to two circles?

$P$ is a point outside a circle whose distance from the centre is $13 cm$. A secant drawn from $P$ intersect the circle at $A$ and $B$ in such a way that $PA=9 cm$ and $AB=7 cm$. What is the radius of the circle?

$PQ$ and $PR$ are two equal chords of a circle with centre $O$. If $\angle QPR=72°$, what is the value of $\angle OPQ$?

The number of equidistant points from the sides which lie on the same plane with the triangle is _____.

In $△ABC, D$ and $E$ are two points on $AB$ and $AC$ and $DE\parallel BC$. If $AD=x, DB=x+2, AE=x-2, EC=x-1,$ then what is the value of $x$?

In $△ABC,$ the points $D$ and $E$ lie on $AB$ and $AC$ in such a way that $AD:AB=1:2$ and $AE:AC=1:2$. If $BC=6 cm$, what is the length of $DE$?

If the ratio of the areas of two similar triangles $9:16$ then what is the ratio of their corresponding sides?

$BD$ is the diameter of the circumcircle $ABC$. $AD$ produced and $BC$ produced meet at $E$. If $\angle DBC=30°$ and $\angle DEC=20°$, then find the angles of $∆ABC$.