Basic Proportionality Theorem (Thales Theorem)

In the adjoining figure , medians $AD$ and $BE$ of $△ABC$ meet at the point $G,$ and $DF$  is drawn parallel to $BE,$ Prove that $EF=FC$ and $AG : GD=2 : 1$

In the adjoining quadrilateral $ABCD$, $AB=AD.$ The bisectors of $\angle BAC$ and $\angle CAD$ meet the sides $BC$ and $DC$ at the points $E$ and $F$ respectively. Prove that $EF$ is parallel to $BD.$

In a $\Delta ABC\mathit{,}D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that,  $AD=5.7 \mathrm{cm}, BD=9.5 cm, AE=3.3 cm$ and $AC=8.8 \mathrm{cm}$ Is $DE\parallel BC$ ? Justify your answer.

In a $\Delta ABC\mathit{,}D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE \Vert BC$. If $AD=2.4 \mathrm{cm}, AE=3.2 \mathrm{cm},DE=2 \mathrm{cm}$ and $BC=5 \mathrm{cm},$ find $BD$ and $CE$

In the adjoining figure, $ABC$ is a triangle right-angled at $B$ and $BD\perp AC.$ If $AD=4 cm$ and $CD=5 cm$, find $BD$ and $AB.$

In the adjoining figure, line segment $DF$ intersect the side $AC$ of triangle $ABC$ at the point $E$ such that $E$ is mid-point of $CA$ and $\angle AEF=\angle AFE$. Prove that $\frac{BD}{CD}=\frac{BF}{CE}$.

If the diagonals of a quadrilateral divide each other proportionally, prove that it is a trapezium.

$D$ d and $E$ E are respectively the points on the sides $AB$ and $AC$ of a $△ABC$ such that $AD=2 cm, BD=3 cm, BC=7.5 cm$ and $DE\parallel BC$. Then the length of $DE$ is

In the adjoining figure, $MN\parallel QR$ . If $PN=3.6 cm, NR=2.4 cm$ and $PQ=5 cm$, then $PM$ is

In the adjoining figure, $PQ\parallel CA$ and lengths are given in centimetres. The length of $BC$ is

In the adjoining figure, $DE\parallel BC$ and all measurements are in centimetres. The length of $AE$ is

In the adjoining figure, $\angle D=\angle E$ and $\frac{AD}{DB}=\frac{AE}{EC}$. Prove that $BAC$ is an isosceles triangle.

In the adjoining figure, $CD\parallel LA \mathrm{and} DE\parallel AC$. If $BE=4 \mathrm{cm}, EC=2 \mathrm{cm}$ and $CL=k \mathrm{cm}$, find $k$.

In figure given below, $\mathrm{AB}\parallel \mathrm{DE} \mathrm{and} \mathrm{BD}\parallel \mathrm{EF}$. Prove that ${\mathrm{DC}}^2=\mathrm{CF}\times \mathrm{AC}$.

In figure given below, $\mathrm{DE}\parallel \mathrm{BC} \mathrm{and} \mathrm{BD}=\mathrm{CE}$. Prove that $\mathrm{ABC}$ is an isosceles triangle.

$X \mathrm{and} Y$ are points on the sides $AB \mathrm{and} AC$ respectively of a triangle ABC such that $\frac{AX}{AB}=\frac{1}{4}, AY=2 \mathrm{cm}$ and $YC=6 \mathrm{cm}$. Find whether $XY\parallel BC$ or not.
 

$\mathrm{A}$ and $\mathrm{B}$ are respectively the points on the sides $\mathrm{PQ}$ and $\mathrm{PR}$ of a triangle $\mathrm{PQR}$ such that $\mathrm{PQ}=12.5 \mathrm{cm}, \mathrm{PA}=5 \mathrm{cm}, \mathrm{BR}=6 \mathrm{cm}$ and $\mathrm{PB}=4 \mathrm{cm}$. Is $\mathrm{AB}\parallel \mathrm{QR}$? Give reasons for your answer.

$\mathrm{E}\mathrm{}$ and $\mathrm{F}$ are points on the sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{R}$ respectively of a $∆\mathrm{P}\mathrm{Q}\mathrm{R}$. For each of the following cases, state whether $\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$:

$\mathrm{PQ}=1.28 \mathrm{cm}, \mathrm{PR}=2.56 \mathrm{cm}, \mathrm{PE}=0.18 \mathrm{cm}$ and $\mathrm{PF}=0.36 \mathrm{cm}$

$\mathrm{E}\mathrm{}$ and $\mathrm{F}$ are points on the sides $\mathrm{P}\mathrm{Q}$ and $\mathrm{P}\mathrm{R}$ respectively of a $∆\mathrm{P}\mathrm{Q}\mathrm{R}$. For the following case, state whether $\mathrm{E}\mathrm{F}\parallel \mathrm{Q}\mathrm{R}$:
 $\mathrm{P}\mathrm{E}=4 \mathrm{c}\mathrm{m}, \mathrm{Q}\mathrm{E}=4.5 \mathrm{c}\mathrm{m}, \mathrm{P}\mathrm{F}=8 \mathrm{c}\mathrm{m}$ and $\mathrm{R}\mathrm{F}=9 \mathrm{c}\mathrm{m}$

In the figure given below, $D \mathrm{and} E$ are the points on sides $AB \mathrm{and} AC$ respectively. such that $DE\parallel BC$. If $AD=6x-7, DB=4x-3, AE=3x-3, EC=2x-1$.Find the value of $x$.