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.

Short answer

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

Step-by-step solution

Understand the Properties of the Triangle

A right triangle has a hypotenuse, the longest side, labeled as \(h\), and one of the legs of length \(x\). We need to find the area \(A(x)\) in terms of \(x\).

Use the Pythagorean theorem

Since this is a right triangle, using the Pythagorean theorem, \(x^2 + b^2 = h^2\), where \(b\) is the length of the other leg. This can be rearranged to find \(b\): \(b^2 = h^2 - x^2\), so \(b = \sqrt{h^2 - x^2}\).

Use the Formula for the Area of a Triangle

The area of a right triangle is given by \(A = \frac{1}{2} \times \, \text{base} \, \times \, \text{height}\). In this case, the legs of the triangle work as the base \(x\) and height \(b\). Therefore, the area \(A(x)\) is \(\frac{1}{2} \times x \times b\).

Substitute the Expression for \(b\) into the Area Formula

Substitute \(b = \sqrt{h^2 - x^2}\) into the formula for the area: \[ A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \].

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in right triangle geometry, often remembered with the formula: \(a^2 + b^2 = c^2\). This equation relates the squares of the lengths of the two shorter sides, or legs, of a right triangle to the square of the hypotenuse, which is the longest side. In our problem, we have a right triangle with a hypotenuse of length \(h\) and one leg of length \(x\). By applying the Pythagorean theorem, we can solve for the length of the other leg \(b\).

Using the theorem, \(x^2 + b^2 = h^2\). We can rearrange this equation to find \(b^2 = h^2 - x^2\). To find \(b\), take the square root of both sides, giving \(b = \sqrt{h^2 - x^2}\).

This important step allows us to express the unknown side, \(b\), in terms of known quantities, thus bridging the gap to solving for the area of the triangle. Understanding and applying the Pythagorean theorem is crucial for tackling geometric problems related to right triangles.

triangle geometry

In the study of triangle geometry, each type of triangle has specific properties that define it. A right triangle has one 90-degree angle, which simplifies calculations by allowing the use of trigonometric relationships. In the context of our exercise, one of the primary properties is that the side lengths are related through the Pythagorean theorem, as we previously discussed.

Right triangles make it easy to calculate dimensions and areas using their known angle and side properties. In our problem, with the hypotenuse \(h\) already known, and one side \(x\) given, you can easily find the other side \(b\) with the Pythagorean theorem. The symmetry and ratios in right triangles are often utilized across various applications, from architecture to physics.

Understanding the basics of triangle geometry helps in efficiently applying the right theorems and formulas, as seen when dissecting our specific problem. Familiarity with these geometric principles is handy not just for solving textbook problems, but for practical real-world applications.

area formula

The area formula for a triangle is a basic yet powerful tool in geometry. Specifically, the area \(A\) of any triangle is given by the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). For a right triangle, its two legs often serve as the base and height since they meet at a 90-degree angle.

In our context, \(x\) can be the base while \(b\) is the height. Thus, the area \(A(x)\) becomes \(\frac{1}{2} \times x \times b\). With \(b = \sqrt{h^2 - x^2}\) from the Pythagorean theorem, substituting gives us:

• \(A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2}\)

This expression provides the area of the triangle in terms of the length of one leg, \(x\), and the hypotenuse, \(h\).

Calculating the area involves straightforward multiplication and understanding how to manipulate expressions with roots. Mastering this formula and knowing how to tailor it to different triangle problems is a vital skill. Simplifying such expressions is useful for mathematics and helps sharpen analytical skills in everyday scenario solving.