- Identify the triangle as a special 45-45-90 right triangle where both legs equal 12 cm.
- Use the Pythagorean theorem or the specific property \(c = a\sqrt{2}\) to relate the leg length to the hypotenuse.
- Calculate the hypotenuse by multiplying the leg length by \(\sqrt{2}\).
- Conclude that the length of the hypotenuse is \(\boxed{12\sqrt{2} \text{ cm}}\).
Explanation
- Analyze the triangle properties
A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is a special type of right-angled triangle where the two non-right angles are equal, meaning the sides adjacent to the right angle (the legs) are also equal. Given that each leg of this triangle measures \(12 \text{ cm}\), we have two legs, \(a = 12 \text{ cm}\) and \(b = 12 \text{ cm}\). We need to find the length of the hypotenuse, which we will denote as \(c\).
- Apply the Pythagorean theorem or triangle properties
In any right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the Pythagorean theorem states that \(a^2 + b^2 = c^2\). For a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle specifically, because the legs are equal (\(a=b\)), the relationship simplifies to \(a^2 + a^2 = c^2\), which means \(2a^2 = c^2\). Taking the square root of both sides, we get the formula for the hypotenuse: \(c = a\sqrt{2}\).
- Calculate the hypotenuse length
Substitute the given leg length of \(12 \text{ cm}\) into the formula \(c = a\sqrt{2}\):
\[c = 12\sqrt{2} \text{ cm}\]
Alternatively, using the Pythagorean theorem:
\[12^2 + 12^2 = c^2\]
\[144 + 144 = c^2\]
\[288 = c^2\]
\[c = \sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2} \text{ cm}\].
- State the final answer
The length of the hypotenuse is \(12\sqrt{2} \text{ cm}\). This matches the fourth option provided in the problem. The final answer is \(\boxed{12\sqrt{2} \text{ cm}}\).
Examples
In carpentry and construction, when building a perfect square corner or a structural brace, you often use the properties of a 45-45-90 triangle. For example, if you are cutting a piece of wood for a diagonal support beam in a square frame where the side length is 12 cm, you can quickly determine the exact length of the support needed by calculating the hypotenuse as \(12\sqrt{2} \approx 16.97 \text{ cm}\). This ensures the structural integrity and symmetry of your build.