In the triangle ABC, AD and BE are two medians and DF parallel to BE, meets AC at the point F. If the length of the side AC is $8 cm$., then let us write the length of the side CF.

The line parallel to $BC$ of $∆ABC$ meets $AB$ and $AC$ at $P$ and $Q$ respectively. If $AP=4 cm$, $QC=9 cm$ and $PB=AQ$, then find the length of $PB$.

Prove that, if a perpendicular is drawn on the hypotenuse from the right angular point of a right-angled triangle, two triangles so formed on the two sides of the perpendicular are each similar to the original triangle and also similar to each other.

In the given figure $DE \left|\right| BC$ and $\frac{AD}{DB}=\frac{3}{5}$ if $AC=16$ units then the value of $AE$ will be

In $∆\mathrm{ABC}, \mathrm{LM}\parallel \mathrm{BC}$ and $\frac{\mathrm{AL}}{\mathrm{LB}}=\frac{2}{3}, \mathrm{AM}=5 \mathrm{cm}$, find $\mathrm{AC}$

In $∆ABC, DE\parallel BC,$ If $AD=5 cm, BD=7 cm$ and $AC=18 cm,$ If the length of $AE=k$, then find the value of $k$.

In Fig 2, $DE\parallel AC$ and $DC\parallel AP$. Prove that $\frac{BE}{EC}=\frac{BC}{CP}$

In $△ABC,$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$. If $AP=8 \mathrm{cm}, CQ=9 \mathrm{cm}$ and $AQ=2BP$, then what is the value of $BP$?

In $∆ABC, O$ is a point on the median $AD. BO$ and $CO$ produced meet $AC$ and $AB$ at $G \& F$. Prove that $FG\parallel BC$.

Two parallel straight lines intersect three concurrent straight lines at $A,B,C$ and $X,Y,Z$. Prove that $AB:BC=XY:YZ$.

In a $∆ABC$, a line parallel to $BC$ intersects $AB$ and $AC$ at $P$ and $Q$. If $AQ=3AP$, then $BP:CQ$ is _____.

In trapezium $ABCD, AB\parallel CD$; The diagonals $AC \& BD$ meet at $O$. Prove that $AO\times OD=BO\times OC$.

In $△ABC,$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$. If $AB=3PB$, then find $PQ:BC$.

In trapezium $ABCD$, $AB$ and $DC$ are parallel. A straight line parallel to $AB$ meets $AD$ and $BC$ at $E$ and $F$. Prove that $DE:EA=CF:FB$.

In the figure,$\mathrm{ABC}$ is a triangle in which $\mathrm{AB}=\mathrm{AC}$ . Points$\mathrm{D}$ and $\mathrm{E}$ are points on the side $\mathrm{AB}$ and $\mathrm{AC}$ respectively such that $\mathrm{AD}=\mathrm{AE}$. Show that the points $\mathrm{B}, \mathrm{C}, \mathrm{E}$ and $\mathrm{D}$ lie on a same circle.

In $△ ABC, DE\parallel DE\parallel BC$. If $AD=5 \mathrm{cm}, BD=7 \mathrm{cm}$ and $AC=18 \mathrm{cm}$, find the length of $AE$ in centimrter.

In $∆ABC$ the line parallel to $BC$ meets $AB$ and $AC$ at $P$ and $Q$ and $\frac{AP}{PB}=\frac{2}{5}$. If $AC=28 \mathrm{cm}$, then length of $AQ$ is 

In $∆ABC$ the line parallel to $BC$ meets $AB$ and $AC$ at $X$ and $Y$ and $AX:XB=3:1$. If $AY=6.6 \mathrm{cm}$, then length of $AC$ is