The Hypotenuse Of A 45-45-90 Triangle Measures 18 Cm.

The Hypotenuse of a 45-45-90 Triangle Measures 18 cm: Unpacking the Geometry

The seemingly simple statement – "the hypotenuse of a 45-45-90 triangle measures 18 cm" – opens a door to a world of geometric exploration. This seemingly straightforward problem allows us to delve into the properties of isosceles right-angled triangles, apply the Pythagorean theorem, and understand the relationship between angles and side lengths in right-angled triangles. This article will comprehensively dissect this problem, offering a detailed explanation suitable for various levels of mathematical understanding.

Understanding 45-45-90 Triangles

Before we dive into the specifics of our 18 cm hypotenuse, let's establish a firm understanding of 45-45-90 triangles. These triangles, also known as isosceles right triangles, are characterized by:

• Two equal angles: The angles other than the right angle (90 degrees) are both 45 degrees. This is where the name "45-45-90" originates.
• Two equal sides: The sides opposite the 45-degree angles are congruent (equal in length). These are often referred to as the legs of the triangle.
• Hypotenuse: The side opposite the right angle is the hypotenuse, always the longest side in a right-angled triangle.

This inherent symmetry is a key feature that simplifies calculations. The ratio of the sides in a 45-45-90 triangle is always 1:1:√2. This means that if the legs have length 'x', the hypotenuse has length x√2.

Applying the Pythagorean Theorem

The Pythagorean theorem is fundamental to understanding right-angled triangles. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). Mathematically:

a² + b² = c²

Where:

• 'a' and 'b' are the lengths of the legs.
• 'c' is the length of the hypotenuse.

In our case, we know the hypotenuse (c = 18 cm). We can use the Pythagorean theorem to find the lengths of the legs. Since it's a 45-45-90 triangle, we know a = b. Therefore, the equation becomes:

a² + a² = 18²

2a² = 324

a² = 162

a = √162

a = 9√2 cm

Therefore, both legs of the triangle have a length of 9√2 cm.

Solving Using the Ratio of Sides

Alternatively, we can leverage the 1:1:√2 ratio inherent to 45-45-90 triangles. We know the hypotenuse is 18 cm, which corresponds to the √2 portion of the ratio. To find the length of the legs (represented by '1' in the ratio), we can set up a proportion:

1 / √2 = x / 18

Solving for x (the length of each leg):

x = 18 / √2

To rationalize the denominator (eliminate the square root from the bottom), we multiply the numerator and denominator by √2:

x = (18√2) / (√2 * √2)

x = (18√2) / 2

x = 9√2 cm

This confirms our earlier result that each leg of the triangle measures 9√2 cm.

Approximating the Lengths

The value 9√2 is an exact value. However, for practical applications, we might want an approximate decimal value. Using a calculator:

√2 ≈ 1.414

Therefore:

9√2 ≈ 9 * 1.414 ≈ 12.726 cm

So, the legs are approximately 12.726 cm long.

Further Applications and Extensions

The problem of a 45-45-90 triangle with a hypotenuse of 18 cm provides a springboard for exploring more complex geometric concepts:

Area Calculation

Once we know the lengths of the legs, we can easily calculate the area of the triangle using the formula:

Area = (1/2) * base * height

In our case, the base and height are both 9√2 cm, so:

Area = (1/2) * (9√2) * (9√2) = (1/2) * 162 = 81 cm²

The area of the triangle is 81 cm².

Trigonometry

45-45-90 triangles offer simple scenarios for introducing trigonometric functions. Since the legs are equal, we have:

sin(45°) = cos(45°) = (9√2) / 18 = 1/√2 = √2/2

tan(45°) = 1

These relationships are fundamental in trigonometry and are often used in solving more complex problems involving angles and side lengths.

Three-Dimensional Applications

The principles applied here extend to three-dimensional geometry. Imagine a cube; each of its faces is a square, which can be divided into two 45-45-90 triangles. Understanding the properties of these triangles is crucial for calculations involving volume, surface area, and diagonals of cubes and other three-dimensional shapes.

Conclusion: Beyond the Numbers

The seemingly simple problem of determining the leg lengths of a 45-45-90 triangle with an 18 cm hypotenuse provides a rich opportunity to explore foundational geometric concepts. By applying the Pythagorean theorem and understanding the unique ratio of sides in this type of triangle, we can accurately calculate the lengths of the legs and the area of the triangle. Moreover, this problem serves as a stepping stone for understanding more advanced concepts in geometry and trigonometry, showcasing the interconnectedness of mathematical principles and their practical applications in various fields. The ability to solve this seemingly simple problem demonstrates a grasp of fundamental geometric principles and opens doors to tackling more complex geometric challenges. From simple area calculations to three-dimensional applications, the knowledge gained through solving this problem provides a strong foundation for further mathematical exploration. Remember, even seemingly simple problems can reveal profound insights into the elegant world of mathematics.