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:

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:

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 $\triangle ABC$?

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 $\triangle CBF$.

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 $\triangle ABC$?