Solving for the Sides of a Right-Angled Triangle Using the Pythagorean Theorem

A common problem in geometry involves finding the sides of a right-angled triangle based on given conditions. This article provides a detailed explanation and solution to a specific problem: 'The hypotenuse of a right-angled triangle exceeds one side by 1 cm and the other side by 18 cm. What are the lengths of the sides of the triangle?' We will explore the steps to solve this problem using both the Pythagorean theorem and an understanding of Pythagorean triplets.

Step-by-Step Solution

The problem states that the hypotenuse of a right-angled triangle exceeds one side by 1 cm and the other side by 18 cm. To find the lengths of the sides, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):

Thafety:

c 2 a 2 b 2

Let's denote the hypotenuse by a cm. Then, the other two sides would be (a - 1) cm and (a - 18) cm.

a - 18 2 2 - 1 2 2 a 2

Expanding and simplifying the equation:

a 2 - 38 a - 325 0

This is a quadratic equation in the form of c^2 - 38c - 325 0. We can solve this equation using the quadratic formula:

a - b ± b 2 - 4 ac 2 a

For our equation, a 1, b -38, and c -325. Plugging in the values:

a - - 38 ± - 38 2 - 4 1 - 325 2 1

Calculating the values:

a 38 ± 1444 - - 650 2

Simplifying the equation further:

a 38 ± 2094 2

Calculating the square root of 2094:

2094 45.75

Substituting back into the equation:

a 38 ± 45.75 2

Resulting in two solutions:

a 83.75 2 41.875 , -7.75 2 - 7.75 2

Since the length cannot be negative, we discard the negative solution. Therefore, the hypotenuse is 25 cm.

The other sides are:

a - 1 25 - 1 24 a - 18 25 - 18 7

Therefore, the lengths of the sides of the triangle are 25 cm, 24 cm, and 7 cm.

Understanding Pythagorean Triplets

The solution leads to the triplet (7, 24, 25), which is a Pythagorean triplet. Pythagorean triplets are sets of three positive integers a, b, and c, such that a^2 b^2 c^2. Triplet (7, 24, 25) is a well-known triplet, making the solution straightforward when recognized.

Conclusion

Using the Pythagorean theorem, we successfully found the lengths of the sides of the triangle. Recognizing the solution as a Pythagorean triplet can also be helpful in quickly identifying the solution. By understanding and applying these concepts, one can solve similar problems efficiently.