Question : The midpoints of sides AB and AC of the triangle ABC are, respectively, X and Y. If (BC + XY) = 12 units, then the value of (BC – XY) is:

Option 1: 2 units

Option 2: 6 units

Option 3: 8 units

Option 4: 4 units

Question : D, E, and F are the midpoints of the sides BC, CA, and AB, respectively of a $\triangle ABC$. Then the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ is:

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

Option 2: $\frac{1}{4}$

Option 3: $\frac{1}{8}$

Option 4: $\frac{1}{16}$

Question : ABC is an equilateral triangle. If the area of the triangle is $36 \sqrt{3}$, then what is the radius of the circle circumscribing the $\triangle ABC$?

Option 1: $2 \sqrt{3}$

Option 2: $3 \sqrt{3}$

Option 3: $4 \sqrt{3}$

Option 4: $6 \sqrt{3}$

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 : ABC is an isosceles right-angle triangle. $\angle ABC = 90 ^{\circ}$ and AB = 12 cm. What is the ratio of the radius of the circle inscribed in it to the radius of the circle circumscribing $\triangle ABC$?

Option 1: $6–\sqrt{2}: 3 \sqrt{2}$

Option 2: $2–\sqrt{2}: \sqrt{2}$

Option 3: $6–3 \sqrt{2}: 1 \sqrt{2}$

Option 4: $6–3 \sqrt{2}: 6 \sqrt{2}$