Introduction to Similar Triangles and Perimeter Ratios

Understanding the properties of similar triangles and how to use given ratios to find areas is crucial in geometry and practical applications. This article focuses on finding the areas of two similar triangles where the ratio of their perimeters is 4:3, and the sum of their areas is 130 cm2.

Understanding the Ratio of Perimeters

The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding side lengths. If the ratio of the perimeters is 4:3, then the ratio of the corresponding side lengths is also 4:3. This ratio is fundamental in determining the areas of the triangles, as we will see.

Finding the Ratio of Areas

The areas of similar triangles are proportional to the square of the ratio of their corresponding side lengths. Therefore, if the ratio of the side lengths is 4:3, the ratio of the areas will be:

(left(frac{4}{3}right)^2 frac{16}{9})

Setting Up the Equations

Let the areas of the two triangles be (A_1) and (A_2). According to the ratio of the areas:

(frac{A_1}{A_2} frac{16}{9})

We can express (A_1) in terms of (A_2):

(A_1 frac{16}{9} A_2)

Using the sum of the areas:

(A_1 A_2 130) cm2

Substituting (A_1) in terms of (A_2):

(frac{16}{9} A_2 A_2 130)

To combine the terms, express (A_2) with a common denominator:

(frac{16}{9} A_2 frac{9}{9} A_2 130)

(frac{25}{9} A_2 130)

Solving for (A_2):

Multiply both sides by 9:

(25 A_2 1170)

Divide by 25:

(A_2 frac{1170}{25} 46.8) cm2

Now, substituting (A_2) back to find (A_1):

(A_1 frac{16}{9} times 46.8 frac{748.8}{9} 83.2) cm2

Final Areas

The areas of the two triangles are:

- (A_1 83.2) cm2 - (A_2 46.8) cm2

Thus, the areas of the triangles are 83.2 cm2 and 46.8 cm2.

This solution demonstrates the powerful application of the properties of similar triangles in solving real-world problems involving perimeter and area computations.