Question

Let $G$ be the centroid of triangle $ABC$. If triangle $ABG$ is an equilateral triangle with a side length of $2$, then find the perimeter of triangle $ABC$.

a. $4$
b. $6$
c. $8$
d. $10$

Answer 1

The perimeter of triangle ABC is closest to 6 units when triangle ABG is an equilateral triangle with each side length equal to 2, making option b the correct answer.

Step-by-step explanation:

To calculate the perimeter of triangle ABC when triangle ABG is equilateral and centroid G divides medians in a 2:1 ratio, we can use the fact that the medians of a triangle are equally divided by its centroid. Since G is the centroid and ABG is equilateral with each side equal to 2, then AG equals two-thirds of AB. This means AB = 3 units because AG = (2/3)(AB) = 2. Thereby, ABC is isosceles with AB = BC = 3, and the third side AC must be smaller than 3. However, this side AC must have a length such that when added with the lengths of AB and BC it enables the centroid G to be within an equilateral triangle ABG of side 2. Finally, we can determine that the perimeter is the sum of AB (3 units), BC (3 units), and AC (less than 3 units), which is strictly greater than 6 units but less than 9 units. Therefore, the perimeter cannot be exactly 8 or 10 units, which leaves us with a perimeter closest to 6 units, which is option b.