Determine the Sides of a Right Triangle Using Algebraic Relationships

In this article, we will explore a right triangle with specific side lengths described through algebraic relationships. We will use the Pythagorean theorem and algebraic equations to determine the lengths of the sides. This problem is a classic application of both algebra and geometry, and it can help readers understand the interplay between these fields.

Introduction to the Problem

Let's consider a right triangle where the relationships between the sides are given as follows:

The hypotenuse is 6 cm more than twice the smallest side. The length of the third side (second side) is 2 cm less than the hypotenuse.

Defining the Variables

To solve this problem, we will denote the sides of the triangle as follows:

a: the length of the smallest side. b: the length of the second side. c: the length of the hypotenuse.

Setting Up the Equations

From the problem, we can set up the following relationships:

c 2a 6 b c - 2

Solving for the Sides

First, let's express b in terms of a:

b (2a 6) - 2 2a 4

Now, substituting the expressions for b and c into the Pythagorean theorem (a2 b2 c2):

a2 (2a 4)2 (2a 6)2

Expanding both sides:

a2 (4a2 16a 16) 4a2 24a 36

Simplifying the left side:

a2 4a2 16a 16 5a2 16a 16

Setting the two sides equal:

5a2 16a 16 - 4a2 - 24a - 36 0

Rearranging the equation:

a2 - 8a - 20 0

Solving this quadratic equation using the quadratic formula:

a [-b ± √(b2 - 4ac)] / (2a)

Here, a 1, b -8, and c -20:

a [8 ± √((-8)2 - 4*1*(-20))] / (2*1)

a (8 ± √(64 80)) / 2

a (8 ± √144) / 2

a (8 ± 12) / 2

This gives us two possible values for a:

a (8 12) / 2 20 / 2 10 a (8 - 12) / 2 -4 / 2 -2

Since side lengths cannot be negative, we take a 10 cm.

Calculating the Other Sides

Now, we can find b and c:

b 2a 4 2*10 4 24 cm c 2a 6 2*10 6 26 cm

Final Answer

The lengths of the sides of the triangle are:

Smallest side: 10 cm Second side: 24 cm Hypotenuse: 26 cm

This solution illustrates the application of algebraic relationships and the Pythagorean theorem to solve geometric problems. By understanding these concepts, readers can tackle similar problems effectively.

Conclusion

Understanding how to apply algebraic relationships in geometric problems is a valuable skill in mathematics. The steps outlined in this article provide a clear and systematic approach to solving such problems. Whether you are studying for math exams or simply exploring mathematical concepts, this method can be a helpful tool in your problem-solving arsenal.