Question : In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$, such that $DE$ II $BC$. If $AD=x+3$, $DB =2 x-3$, $A E=x+1$ and $EC=2 x-2$, then the value of $x$ is:

Option 1: $\frac{4}{5}$

Option 2: $\frac{1}{2}$

Option 3: $\frac{3}{5}$

Option 4: $\frac{1}{5}$

Question : In $\triangle ABC$, D and E are points on the sides AB and AC, respectively, such that DE || BC. If AD = 5 cm, DB = 9 cm, AE = 4 cm, and BC = 15.4 cm, then the sum of the lengths of DE and EC (in cm) is:

Option 1: 11.6

Option 2: 10.8

Option 3: 13.4

Option 4: 12.7

Question : In $\triangle \text{ABC}, \mathrm{DE} \| \mathrm{BC}$ and $\frac{\text{AD}}{\text{DB}}=\frac{4}{5}$. If $\mathrm{DE}=12 \mathrm{~cm}$, find the length of $\mathrm{BC}$.

Option 1: 48 cm

Option 2: 12 cm

Option 3: 30 cm

Option 4: 27 cm

Question : In $\triangle$ABC, the bisector of $\angle$BAC intersects BC at D and the circumcircle of $\triangle$ABC at E. If AB : AD = 3 : 5, then AE : AC is:

Option 1: 5 : 3

Option 2: 3 : 2

Option 3: 2 : 3

Option 4: 3 : 5

Question : In a $\triangle ABC$, if $\angle A=90^{\circ}, AC=5 \mathrm{~cm}, BC=9 \mathrm{~cm}$ and in $\triangle PQR, \angle P=90^{\circ}, PR=3 \mathrm{~cm}, QR=8$ $\mathrm{cm}$, then:

Option 1: $\triangle ABC \cong \triangle PQR$

Option 2: $ar(\triangle ABC)\neq ar(\triangle PQR)$

Option 3: $ar(\triangle ABC) \leq ar(\triangle PQR)$

Option 4: $ar(\triangle ABC)=ar(\triangle PQR)$