Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

A right triangle has a fixed hypotenuse of length \(h\) and one leg that has length \(x\). Find a formula for the area \(A(x)\) of the triangle.

Step-by-Step Solution

Answer

The formula for the area is \( A(x) = \frac{1}{2} x \sqrt{h^2 - x^2} \).

Step 1: Recognize the Components of the Triangle

A right triangle has three sides: the hypotenuse, which is fixed and denoted by \( h \), and two legs, one of which is given as \( x \). The other leg is denoted by \( y \).

Step 2: Apply the Pythagorean Theorem

In a right triangle, the Pythagorean theorem states that the sum of the squares of the two legs is equal to the square of the hypotenuse: \[ x^2 + y^2 = h^2 \]From this, you can solve for \( y \): \[ y = \sqrt{h^2 - x^2} \]

Step 3: Calculate the Area of the Triangle

The area \( A \) of a triangle is given by the formula \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]In this right triangle, the legs \( x \) and \( y \) serve as the base and height. Therefore, the area \( A(x) \) is: \[ A(x) = \frac{1}{2} \times x \times y \]Substituting for \( y \) from the Pythagorean theorem, we get:\[ A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \]

Step 4: Write the Final Formula for Area

The formula derived for the area \( A(x) \) of the triangle as a function of \( x \) is:\[ A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \]This formula expresses the area of the right triangle in terms of the leg \( x \) and the hypotenuse \( h \).

Key Concepts

Right Triangle Fundamentals

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees, known as a right angle. There are three sides:

• The longest side, which is opposite the right angle, is called the hypotenuse.
• The other two sides are referred to as the legs.

The unique property of a right triangle is its compliance with the Pythagorean Theorem, which helps us calculate the length of any side if the lengths of the other two are known. This theorem is foundational in many mathematical calculations, especially in geometry and trigonometry.

Triangle Area Formula Explained

Calculating the area of a triangle involves understanding the basic formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] For a right triangle, one leg can serve as the base while the other leg serves as the height. Their perpendicular nature facilitates easy calculation of the area. Hence, if you know the lengths of the legs (say one leg is labeled as \(x\) and the other \(y\)), the area \(A\) is given by:\[ A = \frac{1}{2} \times x \times y \]This formula applies to all types of triangles but is especially practical for right triangles due to their properties.

Understanding the Hypotenuse

The hypotenuse is the longest side of a right triangle and plays a key role in calculations related to this triangle. Using the Pythagorean Theorem:\[ x^2 + y^2 = h^2 \]we can solve for any missing side. Here, \(h\) represents the hypotenuse, which can be calculated if the lengths of the other sides are known. This property is helpful in finding the area of the triangle when one leg is fixed and the other variable, using the equation:\[ A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \]This shows how the hypotenuse provides a relationship between the legs, aiding in the formulation of important geometric calculations, especially concerning area.