In right triangle ΔLMN, which is right angled at M, if ∠N = 45° and the hypotenuse LN = 6√2 cm, then what is the length (in centimetres) of side MN?

Introduction / Context:
This geometry question tests recognition of a special right triangle, the 45 45 90 triangle. In such triangles, the legs are equal and the hypotenuse has a fixed ratio to the legs. Identifying this pattern allows a quick solution without using trigonometric functions or the full Pythagorean theorem every time.

Given Data / Assumptions:

• Triangle ΔLMN is right angled at M.
• Angle N is 45 degrees, so angle L is also 45 degrees.
• Hypotenuse LN has length 6√2 cm.
• We seek the length of the leg MN.

Concept / Approach:
In a 45 45 90 triangle, the two acute angles are equal, and therefore the legs opposite them are equal in length. The ratio of the hypotenuse to each leg is √2. If each leg is of length s, then the hypotenuse is s√2. Knowing the hypotenuse, we can simply divide by √2 to find the leg length. This is faster than repeatedly applying the Pythagorean theorem.

Step-by-Step Solution:
Since ΔLMN is right angled at M and ∠N = 45°, the remaining angle ∠L is also 45°. Thus ΔLMN is an isosceles right triangle of type 45 45 90. Let the length of each leg (LM and MN) be s. Then the hypotenuse LN has length s√2. Given LN = 6√2, set s√2 = 6√2. Divide both sides by √2 to get s = 6. Therefore, side MN has length 6 cm.

Verification / Alternative check:
We can check with the Pythagorean theorem. If MN = 6 and LM = 6, then LN^2 = MN^2 + LM^2 = 6^2 + 6^2 = 36 + 36 = 72. Taking the square root, LN = √72 = √(36 * 2) = 6√2, which matches the given hypotenuse. This confirms that MN = 6 cm is consistent with both the triangle type and the given hypotenuse length.

Why Other Options Are Wrong:
Values such as 3, 4, or 2 would give much smaller hypotenuse lengths when combined with an equal leg, so they would not match 6√2. For example, if MN = 3, the hypotenuse would be 3√2, not 6√2. The option 3√2 is longer than the leg in a 45 45 90 triangle, but it does not correspond to any side given the required hypotenuse length.

Common Pitfalls:
Some learners mistakenly apply the 30 60 90 triangle ratios or mix up which side corresponds to the given hypotenuse. Others square 6√2 incorrectly, leading to miscalculations. Always remember that in a 45 45 90 triangle, leg : hypotenuse = 1 : √2, and check consistency with Pythagoras to avoid such mistakes.

Final Answer:
The length of side MN is 6 cm.