Determining the Sides of a Right-Angled Triangle with a Given Perimeter

In this article, we will explore the process of finding the lengths of the sides of a right-angled triangle given its perimeter and the relationship between its sides and hypotenuse. Specifically, we will solve a problem where the perimeter of a right-angled triangle is 40 meters, and the hypotenuse is 2 units longer than one of the other sides.

Understanding the Problem

Given a right-angled triangle (RAT) with sides (a), (b), and (c) (where (c) is the hypotenuse), the perimeter of the triangle is given as:

[a b c 40 , text{meters}]

Additionally, the hypotenuse (c) is 2 units longer than one of the other sides, say (a). Therefore, we can express the hypotenuse as:

[c a 2]

Solving the Equations

Let's denote the sides of the RAT as (a), (b), and (c) where (c) is the hypotenuse.

Step 1: Perimeter Equation

The perimeter equation is:

[a b c 40]

Substituting (c a 2), we get:

[a b (a 2) 40]

Combining like terms, we have:

[2a b 2 40]

Simplifying further:

[2a b 38]

[b 38 - 2a dots 1]

Step 2: Pythagorean Theorem

According to the Pythagorean theorem, the relationship between the sides is:

[a^2 b^2 c^2]

Substituting (c a 2), we get:

[a^2 b^2 (a 2)^2]

Expanding the right side:

[a^2 b^2 a^2 4a 4]

Subtracting (a^2) from both sides:

[b^2 4a 4]

Substituting equation (1) for (b):

[(38 - 2a)^2 4a 4]

Expanding the left side:

[1444 - 152a 4a^2 4a 4]

Bringing all terms to one side:

[4a^2 - 156a 1440 0]

Dividing the entire equation by 4:

[a^2 - 39a 360 0]

Factoring the quadratic equation:

[(a - 24)(a - 15) 0]

This gives us the solutions:

[a 24 , text{or} , 15]

Since (a) and (b) are lengths and must be positive, let's evaluate each solution:

Case 1: (a 24)

[b 38 - 2a 38 - 2(24) 2]

[c a 2 24 2 26]

This solution is inadmissible because the sum of the sides does not equal 40 meters:

[2 24 26 52 , text{meters}]

Case 2: (a 15)

[b 38 - 2a 38 - 2(15) 8]

[c a 2 15 2 17]

This solution is valid as the sum of the sides equals 40 meters:

[8 15 17 40 , text{meters}]

Conclusion

The sides of the right-angled triangle are 8 cm, 15 cm, and 17 cm. This solution satisfies both the perimeter condition and the relationship between the sides and the hypotenuse.

References

For further reading and understanding of similar problems, you can refer to standard algebra and geometry textbooks.