Question : In $\triangle$ABC, D and E are points on the sides BC and AB, respectively, such that $\angle$ACB = $\angle$ DEB. If AB = 12 cm, BE = 5 cm and BD : CD = 1 : 2, then BC is equal to:

Option 1: $8 \sqrt{3}$ cm

Option 2: $5 \sqrt{5}$ cm

Option 3: $6 \sqrt{5}$ cm

Option 4: $6 \sqrt{3}$ cm

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 : Suppose $\triangle ABC$ be a right-angled triangle where $\angle A=90°$ and $AD\perp BC$. If the area of $\triangle ABC =40$ cm$^{2}$ and $\triangle ACD =10$ cm$^{2}$ and $\overline{AC}=9$ cm, then the length of $BC$ is:

Option 1: 12 cm

Option 2: 18 cm

Option 3: 4 cm

Option 4: 6 cm

Question : In $\triangle ABC$, D and E are points on the sides AB and AC, respectively, such that DE || BC and DE : BC = 6 : 7. (Area of $\triangle {ADE}$ ) : (Area of trapezium BCED) = ?

Option 1: 49 : 13

Option 2: 13 : 36

Option 3: 13 : 49

Option 4: 36 : 13

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