EXERCISE

In the Fig., $DE\left|\right|BC$. Find the length of $AE$.

“Note: This question given in the book seems to have errors and the modified question should be as given below.”

Two chords $AB$ and $CD$ of a circle intersect at a point $P$ outside the circle. Prove that $△PAC~△PDB$ and $PA.PB=PD.PC$

In the Fig., $AB\left|\right|CD\left|\right|EF$. If $AB=6 \mathrm{cm}, CD\mathit{=}x \mathrm{cm}, EF=10 \mathrm{cm}, BD\mathit{=}4 \mathrm{cm}$ and $DE=y \mathrm{cm}$, calculate the values of $x$ and $y$.

In a circle, two chords$AB$and $CD$ intersect at a point $P$ inside the circle. Prove that 

$\left(a\right)\hspace{0.17em}△PAC~△PDB$

$\left(b\right)\hspace{0.17em}PA.PB=PC.PD$

In the Fig., if $AB\perp BC, DC\perp BC$ and $DE\perp AC,$ prove $△CED~△ABC$.

Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB\left|\right|DC$ intersect each other at the point $O$. Using similarity criterion for two triangles, show that $\frac{OA}{OC}=\frac{OB}{OD}$

If $△ABC$ and $△AMP$ are two right triangles, right angled at $B$ and $M$ respectively, such that $\angle MAP=\angle BAC$. Prove that: (a) $△ABC~△AMP$ (b) $\frac{CA}{PA}=\frac{BC}{MP}$.  

In a right triangle$ABC$, right-angled at $B, D$ is a point on hypotenuse such that $BD\perp AC$. If $DP\perp AB$ and $DQ\perp BC$ then prove that (a)$D{Q}^2 =DP\times QC$ (b) $D{P}^2 =DQ\times AP$.