Math Problem Statement

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.

Solution

We are given a triangle ABCABC with centroid GG, where ABC\triangle ABC is similar to GAB\triangle GAB. Additionally, the length of side BC=3BC = 3, and the area of ABC\triangle ABC is xx.

Step 1: Understanding the Similarity

Since ABCGAB\triangle ABC \sim \triangle GAB, the corresponding sides of the triangles are proportional. The centroid GG divides each median in the ratio 2:12:1. Therefore, the ratio of similarity between ABC\triangle ABC and GAB\triangle GAB will also be determined by this same ratio, meaning:

Ratio of similarity=3:1\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 ABC\triangle ABC and GAB\triangle GAB is:

(31)2=9\left( \frac{3}{1} \right)^2 = 9

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

Step 3: Expression of Areas

Let the area of GAB\triangle GAB be yy. Then, from the area ratio we know:

x=9yx = 9y

Thus, the total area of ABC\triangle ABC is xx.

Step 4: Finding the Value of 2x22x^2

We are asked to find the value of 2x22x^2. Given that xx is the area of ABC\triangle ABC, we simply compute:

2x2=2×x22x^2 = 2 \times x^2

Since xx is the area, substitute the value of xx and compute accordingly.

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


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 ABC\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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similarity of Triangles
Centroid of a Triangle
Area of Triangles

Formulas

Ratio of similarity between two triangles = (side length ratio)^2
Area of similar triangles = (side length ratio)^2 * area of the smaller triangle
Centroid divides each median in the ratio 2:1

Theorems

Centroid Theorem
Similarity Theorem

Suitable Grade Level

Grades 10-12