91Ó°ÊÓ

Right triangles, a cornerstone of geometry, are triangles where one of the angles is exactly 90 degrees. This unique property lends several special characteristics:

• They are defined by their three sides: two shorter sides, named the 'legs,' and the longest side, called the hypotenuse.
• The base and height of a right triangle are typically the two legs, which form the right angle.
• The area of a right triangle can be calculated as half the product of its base and height: \(\frac{1}{2} \times \text{base} \times \text{height}\).
• The hypotenuse is always opposite the right angle and is the longest side.

In our problem, we use these properties to relate the area to one side of the triangle and express it in terms of a variable \(x\). Therefore, this concept is pivotal for forming the fundamental equations needed to describe the triangle's dimensions.

The Pythagorean theorem is crucial for solving right triangle problems. Stated simply, for a right triangle:- The square of the hypotenuse is equal to the sum of the squares of the other two sides.Mathematically, this is expressed as:\[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs.
In our example, this theorem helps us to relate the given hypotenuse condition to the sides of the triangle. By substituting expressions derived from the area equation, we find:\[x^2 + \left(\frac{30}{x}\right)^2 = (x+1)^2\]Using algebraic manipulation, these expressions aid in forming a cubic equation, a higher-level algebraic expression that helps further dissect the problem.

A cubic equation is a polynomial equation of the form:\[ax^3 + bx^2 + cx + d = 0\]This type of equation can have up to three real roots. Finding the roots involves understanding polynomial behavior. In this exercise, re-arranging and simplifying the right triangle's properties and Pythagorean geometry ends in the cubic equation:\[2x^3 + x^2 - 900 = 0\]
Such an equation may not be solvable through simple algebra alone, so numerical methods or graphing are often employed. For example, techniques like Newton's method or software tools that perform these calculations can help approximate the solutions when solving real-world problems involving cubic equations.

The Intermediate Value Theorem (IVT) provides powerful insights when working with continuous functions. It states if a function \(f(x)\) is continuous on an interval \([a, b]\), and \(N\) is any number between \(f(a)\) and \(f(b)\), then there is at least one \(c\) in \([a, b]\) such that \(f(c) = N\).
In the context of our problem, IVT helps confirm the existence of a positive root. By evaluating the function \(f(x) = 2x^3 + x^2 - 900\) at specific values within the range, like \(x = 0\) and \(x = 13\):

• \(f(0) = -900\), clearly negative.
• \(f(13)\) is positive.

Thus, by passing from a negative to a positive value, IVT assures us there must be a root within \((0, 13)\), providing a useful check for our solution, thereby guiding us to find the actual root that makes sense in our right triangle context.