- Recognize the 45°-45°-90° triangle as an isosceles right triangle with side ratio \(x : x : x\sqrt{2}\).
- Set up the equation \(x\sqrt{2} = 128\), where \(x\) is the leg length and 128 is the hypotenuse.
- Solve for \(x\) by dividing by \(\sqrt{2}\) and rationalizing the denominator: \(x = \frac{128}{\sqrt{2}} = 64\sqrt{2}\).
- The final answer is \(\boxed{64\sqrt{2}\text{ cm}}\).
Explanation
- Analyze the triangle properties and identify the relationship between the hypotenuse and legs.
In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides follow a specific ratio. Since it is an isosceles right triangle, the two legs are of equal length. Let the length of each leg be denoted by \(x\). The relationship between the hypotenuse \(h\) and the legs is given by the Pythagorean theorem: \(x^2 + x^2 = h^2\), which simplifies to \(2x^2 = h^2\), or \(h = x\sqrt{2}\). Given that the hypotenuse measures \(128\text{ cm}\), we have the equation \(x\sqrt{2} = 128\).
- Calculate the length of the leg.
To find the length of one leg \(x\), we divide the hypotenuse by \(\sqrt{2}\):
\[x = \frac{128}{\sqrt{2}}\]
To simplify this expression, we rationalize the denominator by multiplying both the numerator and the denominator by \(\sqrt{2}\):
\[x = \frac{128}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\]
\[x = \frac{128\sqrt{2}}{2}\]
Dividing 128 by 2, we get:
\[x = 64\sqrt{2}\text{ cm}\]
Thus, the length of one leg of the triangle is \(64\sqrt{2}\text{ cm}\).
Examples
Understanding the relationship between the sides of a 45°-45°-90° triangle is incredibly useful in real-world scenarios, such as in architecture and construction. For example, if you are designing a square plot of land and want to place a diagonal path across it, knowing the length of the diagonal allows you to easily calculate the lengths of the bordering sides of the square. If a diagonal path measures 128 meters, the sides of the square would simply be \(128/\sqrt{2} = 64\sqrt{2}\) meters long. This simple geometric principle helps ensure that your designs are perfectly proportioned and symmetrical.