Question : $\triangle ABC$ is an equilateral triangle with a side of 12 cm and AD is the median. Find the length of GD if G is the centroid of $\triangle ABC$.

Option 1: $6 \sqrt{3}$ cm

Option 2: $3 \sqrt{3} $ cm

Option 3: $4 \sqrt{3} $ cm

Option 4: $2 \sqrt{3}$ cm

Question : In a $\triangle ABC$, AD, BE and CF are the medians from vertices A, B, and C, respectively. The point of intersection of AD, BE and CF is called

Option 1: median point

Option 2: orthocentre

Option 3: centroid

Option 4: incentre

Question : In $\triangle$ABC, D and E are points on AB and AC, respectively, such that DE || BC and DE divide the $\triangle$ABC into two parts of equal areas. The ratio of AD and BD is:

Option 1: $1:1$

Option 2: $1:(\sqrt2-$1)

Option 3: $1:\sqrt2$

Option 4: $1:(\sqrt2+$1)

Question : If D and E are the mid-points of AB and AC respectively of $\triangle$ABC then the ratio of the areas of $\triangle$ADE and square BCED is:

Option 1: 1 : 2

Option 2: 1 : 4

Option 3: 2 : 3

Option 4: 1 : 3

Question : $\triangle$ABC is similar to $\triangle$PQR and AB : PQ = 2 : 3. AD is the median to the side BC in $\triangle$ABC and PS is the median to the side QR in $\triangle$PQR. What is the value of $(\frac{BD}{QS})^2$?

Option 1: $\frac{3}{5}$

Option 2: $\frac{4}{9}$

Option 3: $\frac{2}{3}$

Option 4: $\frac{4}{7}$