In the figure, $\mathrm{ABDC}$is a cyclic quadrilateral with $\mathrm{O}$as centre of the circle. If $\angle \mathrm{BOC}$$= 136°$, find $\angle \mathrm{BDC}$.

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In the given figure$\frac{AO}{OC}=\frac{DO}{OB}=\frac{1}{2}$ and $\mathrm{AB} = 4 \mathrm{cm}$. Find the value of $\mathrm{BC}$.

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In the given figure, $\mathrm{AD}$is the bisector of$\angle \mathrm{BAC}$, $\mathrm{AB} = 6 \mathrm{cm}, \mathrm{AC} = 5 \mathrm{cm}$ and $\mathrm{BD} = 3 \mathrm{cm}$. Find $\mathrm{DC}$. It is given that $\angle \mathrm{ABD} = \angle \mathrm{ACD}$.

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In a $△\mathrm{ABC}$, $\mathrm{AD}$is the bisector of $\angle \mathrm{BAC}$, $\mathrm{AB} = 8 \mathrm{cm}, \mathrm{BD} = 6 \mathrm{cm} \mathrm{and} \mathrm{DC} = 3 \mathrm{cm}$. Find $\mathrm{AC}$. (Given that $\angle \mathrm{ABC} = \angle \mathrm{ACB}$).

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If $\mathrm{ABC}$is a quarter circle and a circle is inscribed in it and if $\mathrm{AB} = 1 \mathrm{cm}$, find radius of smaller circle.

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$\mathrm{ABC}$is an equilateral triangle. Point $\mathrm{D}$is on $\mathrm{AC}$and point $\mathrm{AC}$is on $\mathrm{BC}$, such that $\mathrm{AD} = 2\mathrm{CD}$ and $\mathrm{CE} = \mathrm{EB}$. If we draw perpendiculars from $\mathrm{D}$and $\mathrm{E}$to other two sides and find the sum of the length of two perpen- diculars for each set, that is, for D and E individually and denote them as per ($\mathrm{D}$) and per ($\mathrm{E}$) respectively, then which of the following option will be correct.

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$\mathrm{ABCD}$is a trapezium in which $\mathrm{AB}$is parallel to $\mathrm{DC}, \mathrm{AD} = \mathrm{BC}, \mathrm{AB} = 6 \mathrm{cm}, \mathrm{AB} = \mathrm{EF}$ and $\mathrm{DF} = \mathrm{EC}$. If two lines $\mathrm{AF}$and $\mathrm{BE}$are drawn so that area of $\mathrm{ABEF}$is half of $\mathrm{ABCD}$. Find $\mathrm{DF}/\mathrm{CD}$.

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$\mathrm{OD}, \mathrm{OE}$ and $\mathrm{OF}$are perpendicular bisectors to the three sides of the triangle. What is the relationship between $\mathrm{m}\angle \mathrm{BAC}$ and $\mathrm{m}\angle \mathrm{BOC}?$.

If two equal circles of radius $5 \mathrm{cm}$ have two common tangent $\mathrm{AB}$and $\mathrm{CD}$which touch the circle on $\mathrm{A}, \mathrm{C}$and $\mathrm{B}, \mathrm{D}$respectively shown in the figure. If $\mathrm{CD} = 24 \mathrm{cm}$, find the length of $\mathrm{AB}$?