Question : $D$ and $E$ are points on the sides $AB$ and $AC$ respectively of $\triangle ABC$ such that $DE$ is parallel to $BC$ and $AD: DB = 4:5$, $CD$ and $BE$ intersect each other at $F$. Find the ratio of the areas of $\triangle DEF$ and $\triangle CBF$.

Option 1: $16:25$

Option 2: $16:81$

Option 3: $81:16$

Option 4: $4:9$

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 : D and E are two points on the sides AC and BC, respectively of $\triangle ABC$ such that DE = 18 cm, CE = 5 cm, and $\angle$DEC = 90º. If $ \tan\angle$ABC = 3.6, then AC : CD = ?

Option 1: BC : 2CE

Option 2: 2CE : BC

Option 3: 2BC : CE

Option 4: CE : 2BC

Question : A circle is inscribed in a ΔABC having sides AB = 16 cm, BC = 20 cm, and AC = 24 cm, and sides AB, BC, and AC touch circle at D, E, and F, respectively. The measure of AD is:

Option 1: 10 cm

Option 2: 20 cm

Option 3: 6 cm

Option 4: 14 cm

Question : In $\triangle ABC$, D and E are the midpoints of sides BC and AC, respectively. AD and BE intersect at G at the right angle. If AD = 18 cm and BE=12 cm, then the length of DC (in cm ) is:

Option 1: 10

Option 2: 6

Option 3: 9

Option 4: 8