Introduction

When dealing with geometric problems, the Pythagorean theorem is a fundamental tool in solving for unknown side lengths of a right triangle. In this article, we will walk through a step-by-step process to find the dimensions of a rectangle when given certain conditions. Specifically, we will use the Pythagorean theorem to solve a real-world problem: if one side of a rectangle is 4 cm more than the other, and the diagonal is 20 cm, what are the dimensions of the rectangle?

Setting Up the Equation

Let's denote the length of the shorter side of the rectangle as x cm. Consequently, the length of the longer side will be x 4 cm.

According to the Pythagorean theorem, the relationship between the sides and the diagonal of the rectangle is given by:

diagonal2 side12 side22

In this case, the equation becomes:

202 x2 (x 4)2

Solving the Quadratic Equation

First, we calculate 202:

400 x2 x2 8x 16

Combining like terms:

400 2x2 8x 16

Arranging the equation to set it to zero:

2x2 8x - 384 0

Dividing the entire equation by 2:

x2 4x - 192 0

Solving this quadratic equation using the quadratic formula:

x -b pm sqrt{b2 - 4ac} 2a

where a 1, b 4, and c -192.

Calculating the Discriminant

The discriminant is calculated as follows:

b2 - 4ac 42 - 4 × 1 × -192 16 768 784

Substituting back into the quadratic formula:

x -4 pm sqrt{784} 2 × 1

Calculating sqrt{784}:

sqrt{784} 28

So, x -4 ± 28 / 2

Calculating the two possible values for x:

x 24 / 2 12

x -32 / 2 -16 (not a valid length)

Thus, the shorter side x 12 cm.

Finding the Longer Side

The longer side is:

12 4 16 cm

Conclusion

The dimensions of the rectangle are 12 cm (shorter side) and 16 cm (longer side).

Additional Information

To further verify, we can use the Pythagorean theorem to check:

122 162 144 256 400 202

This confirms our solution is correct.