- Identify the triangle as a 45°-45°-90° isosceles right triangle where the hypotenuse \(h\) is related to the leg \(l\) by \(h = l\sqrt{2}\).
- Substitute the given hypotenuse length of 4 cm into the formula.
- Solve for \(l\) and rationalize the denominator to get \(l = 2\sqrt{2} \text{ cm}\).
- Match the result with option (d), \(\boxed{2\sqrt{2} \text{ cm}}\).
Explanation
- Identify the properties of the triangle and the relevant formula.
A \(45^\circ-45^\circ-90^\circ\) triangle is an isosceles right triangle, meaning its two legs are equal in length. Let the length of each leg be denoted by \(l\). The hypotenuse, denoted by \(h\), is the side opposite the right angle. According to the properties of a 45°-45°-90° triangle or the Pythagorean theorem, the hypotenuse is related to the leg length by the formula \(h = l\sqrt{2}\). We are given that the hypotenuse \(h = 4 \text{ cm}\).
- Calculate the length of the leg.
We substitute the given hypotenuse value \(h = 4 \text{ cm}\) into the relationship \(h = l\sqrt{2}\) to solve for the leg length \(l\):
\[4 = l\sqrt{2}\]
To isolate \(l\), we divide both sides by \(\sqrt{2}\):
\[l = \frac{4}{\sqrt{2}}\]
To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{2}\):
\[l = \frac{4 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{4\sqrt{2}}{2}\]
Simplifying the fraction, we get:
\[l = 2\sqrt{2} \text{ cm}\]
Thus, the length of one leg of the triangle is \(2\sqrt{2} \text{ cm}\). This corresponds to option (d).
Examples
In construction, a carpenter might use a square tool to cut a piece of wood for a roof truss that forms a 45°-45°-90° angle. Knowing the hypotenuse length, they can use the relationship \(l = \frac{h}{\sqrt{2}}\) to precisely calculate the length of the rafters needed to maintain structural integrity and correct angles. For example, if the rafters (hypotenuse) must span 4 meters, each base and side leg length would need to be \(2\sqrt{2} \approx 2.83\) meters long.