Understanding a Right Triangle with Given Hypotenuse and One Side

A right triangle is a triangle that has one right angle (90 degrees). In a right triangle, the side opposite to the right angle is called the hypotenuse, which is the longest side of the triangle. However, sometimes the lengths of the other two sides are not given, and only the hypotenuse and one side are provided. This article will guide you through the process of finding the lengths of the missing sides of a right triangle when given the hypotenuse and one side.

Problem Statement

Given a right triangle with a hypotenuse of 17 units and one side that is 7 units longer than the other side, what are the lengths of these two sides?

Solving the Problem Step by Step

We denote the lengths of the two sides of the right triangle as a and b, where b a 7. According to the Pythagorean theorem:

a2 b2 c2

Here, the hypotenuse c is 17 units. Substituting b a 7 into the equation, we get:

a2 (a 7)2 172

Step-by-Step Calculation

1. Expand the equation:

a2 a2 14a 49 289

2. Combine like terms:

2a2 14a 49 289

3. Subtract 289 from both sides:

2a2 14a 49 - 289 0

4. Simplify the equation:

2a2 14a - 240 0

5. Divide the entire equation by 2:

a2 7a - 120 0

Factoring the Quadratic Equation

To solve the quadratic equation, we need to find two numbers that multiply to -120 and add to 7. These numbers are 15 and -8. Therefore:

(a 15)(a - 8) 0

Solving for a, we have:

a 15 0 or a - 8 0

Thus, a -15 (not valid since lengths cannot be negative) and a 8.

Now, substituting a 8 into the equation b a 7 gives us:

b 8 7 15

Therefore, the lengths of the two sides are:

8 units and 15 units

Additional Information and Calculations

1. **Pythagorean Triple (PPT) Verification**: The hypotenuse c 17 is a prime number, and the sides a 8 and b 15 seem to form a PPT. However, upon closer inspection, it is clear that 17 is not part of the 8-15-17 PPT. But, it can be verified that 82 152 64 225 289 172.

2. **Alternative Solution**: Another method involves solving the quadratic equation directly without expanding:

172 x2 (x 7)2

289 x2 x2 14x 49

289 2x2 14x 49 - 289

2x2 14x - 240 0

This simplifies to the same quadratic equation, yielding the same solution.

Conclusion

In summary, the lengths of the sides of the right triangle are 8 units and 15 units. This problem not only demonstrates the application of the Pythagorean theorem but also involves solving a quadratic equation. Understanding these concepts is crucial for solving similar problems in geometry and trigonometry.