Prove that the tangents drawn at the end points of a chord of a circle makes equal angles with the chord.

In figure, $\mathrm{O}$ is the centre of a circle and $\mathrm{BCD}$ is tangent to it at $\mathrm{C}$. Prove that $\angle \mathrm{BAC}+\angle \mathrm{ACD}=90°$.

If from an external point $\mathrm{B}$ of a circle with centre $\mathrm{O}$, two tangents $\mathrm{BC}$ and $\mathrm{BD}$ are drawn such that $\angle \mathrm{DBC}=120°$. Prove that $\mathrm{BO}=2\mathrm{BC}$.

In figure., $\mathrm{PQ}$ is a chord of length  $8$cm of a circle of radius $5$ cm. The tangents at $\mathrm{P}$ and $\mathrm{Q}$ intersect at a point $\mathrm{T}$. Find the length of tangent $\mathrm{TP}$.

In figure, $\mathrm{XY}$ and $\mathrm{X}\text{'}\mathrm{Y}\text{'}$ are two parallel tangents to a circle with centre $\mathrm{O}$ and another tangent $\mathrm{AB}$ with point of contact $\mathrm{C}$ intersecting $\mathrm{XY} \mathrm{at} \mathrm{A}$ and $\mathrm{X}\text{'}\mathrm{Y}\text{'} \mathrm{at} \mathrm{B}$. Prove that $\angle \mathrm{AOB}=90°$.

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In figure, a circle touches all the four sides of a quadrilateral $\mathrm{ABCD}$ with sides $\mathrm{AB}=6 \mathrm{cm}, \mathrm{BC}=7 \mathrm{cm}$ and $\mathrm{CD}=4 \mathrm{cm}$. Find $\mathrm{AD}$.

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