Geometry

In $\triangle ABC$, $\angle A=50^\circ$. $BO$ and $CO$ are the internal bisectors of $\angle B$ and $\angle C$, meeting at $O$. Find $\angle BOC$.

Two similar triangles have areas of $16\ \text{cm}^2$ and $25\ \text{cm}^2$. If the perimeter of the smaller triangle is $32$ cm, find the perimeter of the larger triangle.

From an external point $P$, a tangent $PT$ of length $12$ cm and a secant $PAB$ are drawn to a circle. If $PA=8$ cm, find the length of the chord $AB$.

The angles of a cyclic quadrilateral $ABCD$ are in the ratio $1:2:3:2$ in that order. Find the largest angle.