Equal Intercept Theorem

In $∆ABC$, segment $PQ$ is parallel to $BC$, segment $PQ$ divides $∆ABC$ in two parts which are equal in areas.

Which of the following alternatives indicate the ratio $\frac{BP}{AB}?$

In the following diagram what is the value of $x ?$

A technician wishes to determine the altitude of a ground-mounted long pole .He uses a light source at point$B$ mounted on a structure of height $1$ meters to shine a beam of light through the tops of both the short and long poles. The height of the short pole is $5$ meters and the distance between the short pole and the long pole is $5$meters. The distance between the short pole and the structure supporting the light source is $2$ meters. We assume that the structure supporting the light source and the two poles are all in the same plane and perpendicular to the ground. Find the altitude $ℎ$of the long pole.

In the above figure, $l\parallel m\parallel n$ and $AB=BC$. If $\mathrm{DE}=2.2 \mathrm{cm}$, find $DF$ in $\mathrm{cm}$.

Two straight lines $l_1$ and $l_2$ intersect three straight lines $MN, KL$ and $GH$ at points $A, B, C$ and $D, E, F$ respectively. If $AB=BC$ and $DE=4.2 \mathrm{cm}$ then $DF=?$

State and prove the Intercept Theorem.

In Fig. 3,$LM \left|\right| BC$ and $MN \left|\right| CD$. Prove that $\frac{\mathrm{AL}}{\mathrm{AB}}=\frac{\mathrm{AN}}{\mathrm{AD}}$ 

In Fig. $13.57, \angle A={90}^0 and AD\perp BC.$ Prove that $A{C}^2=DC\times BC$

In figure, $\angle A={90}^0 \mathrm{and} AD\perp BC.$ Prove that $A{D}^2=BD\times DC$

If a line through vertex of a triangle divides the opposite side in the ratio of other two sides, then the line bisects the angle at the vertex.

$ABCD$ is a parallelogram, $E$ is the midpoint of $AB$ and $F$ is the midpoint of $CD$. $GH$ is any line that intersects $AD, EF$ and $BC$ in $G, P$ and $H$, respectively. Find $PH$ if $GP=7.8 \mathrm{cm}$.

In the above figure, $l\parallel m\parallel n$ and $AB=BC$. If $\mathrm{DE}=2.2 \mathrm{cm}$, find $DF$.

Cotyledons are also called-