In a triangle $\mathrm{ABC}, \mathrm{O}$ is the centre of incircle $\mathrm{PQR}, \angle \mathrm{BAC}=65°,$ $\angle \mathrm{BCA}=75°$. find $\angle \mathrm{ROQ}.$

There are two circles each with radius $5 cm$. Tangent $AB$is $26 cm$. The length of tangent $CD$is:

In the given diagram $\mathrm{ABCD}$ is a cyclic quadrilateral $\angle \mathrm{OCB}=50°$ and $\angle \mathrm{BOC}=110°.$ Find the value of $\angle \mathrm{DAO}.$

ABC and CDE are right-angled triangles. $\angle ABC=\angle CDE=90°$, D lies on AC and E lies on BC. $AB=24 cm, BC=60 cm.$ If $DE=10 cm$, then what is the ratio of $\mathrm{CE}: \mathrm{BE} ?$

In the given diagram $∆\mathrm{ABC}$ is an isosceles right-angled triangle, in which a rectangle is inscribed in such a way that the length of the rectangle in twice of breadth. $\mathrm{Q}$ and $\mathrm{R}$ lie on the hypotenuse and $\mathrm{P}, \mathrm{S}$ lie on the two different smaller sides of the triangle. What is the ratio of the areas of the rectangle and that of triangle?

$\mathrm{ABC}$ is an equilateral triangle in which $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ are the mid-points of the sides $\mathrm{AC}, \mathrm{BC}$ and $\mathrm{AB}$ respectively. What is the ratio of the area of the shaded to the unshaded region in the triangle?

$\mathrm{ABC}$ is an isosceles triangle in which $\mathrm{AB}=\mathrm{AC}$ and$(\angle \mathrm{A})=2(\angle \mathrm{B}), \mathrm{AB}=4 \mathrm{cm}$. What is the ratio of inradius to the circumradius?