Question : AD is the median of $\triangle$ABC. If O is the centroid and AO = 10 cm, then OD is:

Option 1: 5 cm

Option 2: 20 cm

Option 3: 10 cm

Option 4: 30 cm

Question : Let A, B, and C be the mid-points of sides PQ, QR, and PR, respectively, of PQR. If the area of $\triangle$ PQR is 32 cm², then find the area of $\triangle$ ABC.

Option 1: 24 cm²

Option 2: 16 cm²

Option 3: 32 cm²

Option 4: 8 cm²

Question : In $\triangle A B C,\ D$ is the mid-point of $BC$ and $G$ is the centroid. If $G D = 10\ \text{cm} $, the length of $AD$ is_____.

Option 1: 20 cm

Option 2: 30 cm

Option 3: 15 cm

Option 4: 10 cm

Question : $ABC$ is a triangle. $AB = 5$ cm, $AC = \sqrt{41}$ cm and $BC = 8$ cm. $AD$ is perpendicular to $BC$. What is the area (in cm²) of $\triangle ABD$?

Option 1: 12

Option 2: 6

Option 3: 10

Option 4: 20

Question : The height of an equilateral triangle is $7 \sqrt{3}$ cm. What is the area of this equilateral triangle?

Option 1: $36 \sqrt{3}$ cm²

Option 2: $25 \sqrt{3}$ cm²

Option 3: $49 \sqrt{3}$ cm²

Option 4: $32 \sqrt{3}$ cm²