Introduction to Right Triangle Analysis

In the realm of Euclidean geometry, the right triangle plays a fundamental role. This article delves into a specific case where the hypotenuse of a right triangle is 2 centimeters more than the longer side, and the shorter side is 7 centimeters less than the longer side. The goal is to determine the lengths of the sides of such a triangle and explore the use of the Pythagorean theorem in solving related geometric problems.

Setting Up the Problem

Let's introduce the variables used to represent the sides of the right triangle:

Longer side: Denoted by x (centimeters) Hypotenuse: h x 2 (centimeters) Shorter side: y x - 7 (centimeters)

Using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can set up the following equation:

h^2 x^2 y^2

Applying the Pythagorean Theorem

Substituting the expressions for h and y into the Pythagorean theorem, we get:

(h x 2)^2 x^2 (x - 7)^2

Expanding and simplifying both sides, we obtain:

Left side: (x 2)^2 x^2 4x 4 Right side: x^2 (x - 7)^2 x^2 x^2 - 14x 49

Combining these, the equation becomes:

x^2 4x 4 2x^2 - 14x 49

Bringing all terms to one side, we have:

0 2x^2 - 18x 45 - x^2 - 4x - 4

Further simplifying, we get:

0 x^2 - 18x 45

Solving the Quadratic Equation

To solve the quadratic equation x^2 - 18x 45 0, we use the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

In this case, a 1, b -18, and c 45. Substituting these values into the formula, we get:

x frac{18 pm sqrt{(-18)^2 - 4 cdot 1 cdot 45}}{2 cdot 1}

Simplifying further:

x frac{18 pm sqrt{324 - 180}}{2} x frac{18 pm sqrt{144}}{2}

Thus, we have:

x frac{18 pm 12}{2}

This gives us two possible solutions:

x frac{30}{2} 15 x frac{6}{2} 3

Since x represents the length of the longer side, we discard the value x 3, as it would make the shorter side negative (x - 7 -4). Therefore, x 15 cm is the valid solution.

Determining the Side Lengths

With x 15 cm, we can find the lengths of the other sides:

Longer side: x 15 cm Shorter side: y x - 7 15 - 7 8 cm Hypotenuse: h x 2 15 2 17 cm

Conclusion and Validation

The length of the hypotenuse in this right triangle is 17 cm. This solution satisfies the conditions given in the problem statement and verifies the correctness of the application of the Pythagorean theorem and quadratic equation in solving geometric problems.