Let AB be  a triangle with centroid G and BC = 3. If ABC is similar to GAB, and if area of ΔABC is x. Then find the value of 2x2.

We are given a triangle \( ABC \) with centroid \( G \), where \( \triangle ABC \) is similar to \( \triangle GAB \). Additionally, the length of side \( BC = 3 \), and the area of \( \triangle ABC \) is \( x \).

### Step 1: Understanding the Similarity
Since \( \triangle ABC \sim \triangle GAB \), the corresponding sides of the triangles are proportional. The centroid \( G \) divides each median in the ratio \( 2:1 \). Therefore, the ratio of similarity between \( \triangle ABC \) and \( \triangle GAB \) will also be determined by this same ratio, meaning:

\[
\text{Ratio of similarity} = 3:1
\]

### Step 2: Area Ratio of Similar Triangles
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the area ratio between \( \triangle ABC \) and \( \triangle GAB \) is:

\[
\left( \frac{3}{1} \right)^2 = 9
\]

Thus, the area of \( \triangle ABC \) is 9 times that of \( \triangle GAB \).

### Step 3: Expression of Areas
Let the area of \( \triangle GAB \) be \( y \). Then, from the area ratio we know:

\[
x = 9y
\]

Thus, the total area of \( \triangle ABC \) is \( x \).

### Step 4: Finding the Value of \( 2x^2 \)
We are asked to find the value of \( 2x^2 \). Given that \( x \) is the area of \( \triangle ABC \), we simply compute:

\[
2x^2 = 2 \times x^2
\]

Since \( x \) is the area, substitute the value of \( x \) and compute accordingly.

If you would like further details or clarifications, feel free to ask!

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Here are 5 related questions to expand your understanding:
1. What is the centroid of a triangle, and how is it calculated?
2. How does the ratio of areas change between similar triangles?
3. How is the centroid used in dividing the areas of triangles?
4. Can you find the side lengths of \( \triangle ABC \) if the length of one side is given and the area is known?
5. How does the similarity of triangles help in calculating unknown areas?

**Tip:** In similar triangles, remember that the ratio of the areas is always the square of the ratio of the corresponding sides.

Let AB be a triangle with centroid G and BC = 3. If ABC is similar to GAB, and if area of ΔABC is x. Then find the value of 2x^2.

This problem explores the geometric properties of triangle ABC with centroid G, given that triangle ABC is similar to triangle GAB. Using the properties of similar triangles and the centroid theorem, the solution calculates the value of 2x^2. Suitable for Grades 10-12.

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