$△$ ABC is an acute angled triangle. DE is drawn parallel to BC as shown. Which of the following are always true?

i) $△ABC\sim △$ ADE

ii) $\frac{AD}{BD}=\frac{AE}{EC}$

iii) $DE=\frac{BC}{2}$

Q.

\(\triangle\) ABC is an acute angled triangle. DE is drawn parallel to BC as shown. Which of the following are always true?

i) \(\triangle ABC \sim \triangle\) ADE

ii) \(\frac{AD}{BD} = \frac {AE}{EC}\)

iii) \( DE = \frac {BC}{2}\)

Q.

D and E are points on the sides AB and AC respectively of a $△$ABC such that DE || BC. Which of the following statement is true?

(i) $△$ ADE ~ $△$ ABC

(ii) $\frac{areaof△ADE}{areaof△ABC}$ = $\frac{A{D}^2}{A{B}^2}$

(iii) $\frac{areaof△ADE}{areaof△ABC}$ = $\frac{A{B}^2}{A{C}^2}$

Q.

In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the two following always hold(s) good? (Here [PQR] denotes the area of triangle PQR.)

(I) [BCX] = [BCY]

(II) [ACX] .[ABY] = [AXY] .[ABC]

Q. In the given figure, $DE\parallel BC$.
(i) Prove that $△ADE$ and $△ABC$ are similar.
(ii) Given taht $AD=\frac{1}{2}BD$, calculate $DE$ if $BC=4.5cm$.
(iii) If area of $△ABC=18c{m}^2$, find the area of trapezium $DBCE$.

Q. $ABC$ is a right angled triangle with $\angle ABC={90}^o$. $D$ is any point on $AB$ and $DE$ is perpendicular to $AC$. Prove that :
(i) $△ADE\sim △ACB$.
(ii) If $AC=13cm$, $BC=5cm$ and $AE=4cm$. Find $DE$ and $AD$.
(iii) Find, area of $△ADE:$ area of quadrilateral $BCED$.