In the adjoining figure the angle $\mathrm{BAC}$ and $\angle \mathrm{ADB}$ are right angles. $\mathrm{BA}=5 \mathrm{cm},\mathrm{AD}=3 \mathrm{cm}$ and $\mathrm{BD}=4 \mathrm{cm},$ what is the length of $\mathrm{DC}$ ?

In the given diagram $\mathrm{AB}\Vert \mathrm{CD},$ then which one of the following is true?

The bisector of the exterior $\angle \mathrm{A}$ of $∆\mathrm{ABC}$ intersects the side $\mathrm{BC}$ produced to $\mathrm{D}$. Here $\mathrm{CF}$ is parallel to $\mathrm{AD}.$ Which one of the following is correct?

The diagonal $\mathrm{BD}$ of a quadrilateral $\mathrm{ABCD}$ bisects $\angle \mathrm{B}$ and $\angle \mathrm{D},$ then:

Two right triangles $\mathrm{ABC}$ and $\mathrm{DBC}$ are drawn on the same hypotenuse $\mathrm{BC}$ on the same side of $\mathrm{BC}.$ If $\mathrm{AC}$ and $\mathrm{DB}$ intersects at $\mathrm{P},$ then:

In an equilateral triangle $\mathrm{ABC},$ the side $\mathrm{BC}$ is trisected at $\mathrm{D}.$ Find the value of ${\mathrm{AD}}^2 :$