Solving the Ratio and Area of a Triangular Garden

This tutorial guides you through solving a geometric problem involving the ratio of the base and height of a triangular garden and its area. By following a step-by-step approach, you will understand how to determine the exact dimensions based on the given ratio and area. This method can be applied to similar geometric problems in geometry and optimization.

Problem Statement

A triangular garden has a base to height ratio of 2:3. The area of the garden is 1/12 of a hectare (approximately 1000 square meters, since 1 hectare 10,000 square meters).

Solution Methodology

The problem requires us to find the dimensions of the base and height of the triangular garden given the ratio and area. Here is a detailed step-by-step breakdown:

Step 1: Define the Variables

Let base b cm and height h cm. According to the given ratio, base to height is 2:3.

Step 2: Formulate the Area Equation

The area of a triangle is given by the formula:

Area 1/2 times; base times; height

Using the given area (1/12 hectare) and converting it to square centimeters (since 1 hectare 10,000,000 square centimeters), we get:

1/12 times; 10,000,000 833,333.33 square cm

Step 3: Express the Height in Terms of the Base

From the ratio, we know:

height 3/2 times; base

Substitute the height in the area formula:

833,333.33 1/2 times; b times; (3/2 times; b)

Step 4: Simplify and Solve for the Base

Simplify the equation:

833,333.33 3/4 times; b^2

3,333,333.32 b^2

b 2000 cm (or 20 meters)

Step 5: Solve for the Height

Using the base value, calculate the height:

height 3/2 times; 2000 3000 cm (or 30 meters)

These are the dimensions of the garden.

Verification Through Different Methods

Let's solve the same problem using a different approach:

Method 2: Using Proportions and Algebra

Let the base x and the height y. The area of the triangle is 1/2 times; x times; y 300. The ratio of base to height is 2:3, so y 3/2 times; x. Substitute y in the area equation: 1/2 times; x times; (3/2 times; x) 300 3/4 times; x^2 300 x^2 400 x 20 (or 20 meters) Calculate the height: y 3/2 times; 20 30 (or 30 meters)

Using both methods, we confirm the base is 20 meters and the height is 30 meters. This solution aligns with the original problem statement and area requirement.

Generalization of the Problem

The methodology used in these examples can be generalized to solve similar problems involving ratios and geometrical figures. For any triangle with a base to height ratio of 2:3 and a given area, the steps outlined above can be followed to determine the exact dimensions. This approach enhances problem-solving skills in geometry and can be applied in various real-world scenarios, such as landscape design and construction.

Conclusion

Understanding and solving geometric problems like this not only improves your mathematical skills but also helps in practical applications. The key takeaway is to use the given ratios and area to form equations and then solve them step-by-step. This detailed process ensures accurate and reliable solutions.