Understanding the Hypotenuse of a Right-Angled Triangle with a 3:4 Ratio

When dealing with right-angled triangles, one of the most intriguing aspects is determining the hypotenuse given the ratio of the sides. This article explores a specific case where the sides of the triangle are in the ratio 3:4 and the area is 24 square centimeters. This example will not only help you understand the relationship between the sides of a right-angled triangle but also how to apply the Pythagorean theorem and area formula for such a triangle.

Problem Statement

Given a right-angled triangle with sides in the ratio 3:4 and an area of 24 square centimeters (cm2), the objective is to determine the length of the hypotenuse.

Step-by-Step Solution

Step 1: Define Variables
Let the sides of the triangle be represented as a 3k, b 4k, and c, where c is the hypotenuse. Here, k is the scaling factor.

Step 2: Use the Area Formula
The area of a triangle is given by the formula: [text{Area} frac{1}{2} times text{base} times text{height}] Given the area of the triangle is 24 cm2, we have:

[text{24} frac{1}{2} times 3k times 4k]

Step 3: Simplify the Equation
Simplifying the above equation:

[text{24} frac{1}{2} times 12k^2] [text{24} 6k^2]

Step 4: Solve for k
Rearrange the equation to solve for k2: [text{6k}^2 24] [k^2 4] [k 2]

Step 5: Calculate the Sides
Now, we can calculate the lengths of the sides:

[text{a} 3k 3 times 2 6 , text{cm}] [text{b} 4k 4 times 2 8 , text{cm}]

Step 6: Find the Hypotenuse
Using the Pythagorean theorem to find the hypotenuse c:

[text{c} sqrt{text{a}^2 text{b}^2}] [text{c} sqrt{6^2 8^2}] [text{c} sqrt{36 64}] [text{c} sqrt{100} 10 , text{cm}]

Key Concepts

1. Triangular Ratios
Knowing that the sides are in the ratio 3:4:5, we can easily find the hypotenuse. The sides of this specific right-angled triangle are in the form of 3x, 4x, and 5x, where x is the scaling factor.

2. Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

3. Area of a Triangle
The area of a triangle is calculated as half the product of its base and height.

Conclusion

By applying the Pythagorean theorem and the area formula, we have successfully determined that the hypotenuse of the given right-angled triangle, with sides in the 3:4 ratio and an area of 24 square centimeters, is 10 centimeters.

Understanding such concepts is crucial for tackling more complex problems in geometry and is a fundamental skill in fields such as engineering, architecture, and physics.