In right triangle LMN, the right angle is at vertex M (∠M = 90°). The measure of angle N is 45°. If the hypotenuse LN has length 9√2 cm, then what is the length (in cm) of side MN?

Introduction:
This problem involves a right triangle with a 45° angle, which means it is a special 45°–45°–90° triangle. Such triangles have fixed relationships between the lengths of their sides. Recognizing and applying these relationships makes the problem straightforward.

Given Data / Assumptions:

• Triangle LMN is right angled at M, so ∠M = 90°.
• Angle N is 45°, so angle L is also 45° (sum of angles in a triangle is 180°).
• The hypotenuse is LN and its length is 9√2 cm.
• We must find the length of side MN.

Concept / Approach:
In a 45°–45°–90° triangle, both legs are equal in length. If each leg has length a, then the hypotenuse has length a√2. Hence, given the hypotenuse, we can reverse this relationship: leg length = (hypotenuse) / √2. This eliminates the need for the Pythagorean theorem calculation from scratch.

Step-by-Step Solution:
Because ∠M = 90° and ∠N = 45°, the third angle ∠L is also 45°. Thus triangle LMN is an isosceles right triangle, with MN = ML. Let each leg be of length a. Then hypotenuse LN = a√2. We are given LN = 9√2 cm, so a√2 = 9√2. Divide both sides by √2 to get a = 9 cm. Thus MN = a = 9 cm.

Verification / Alternative check:
Check with Pythagoras. If MN = 9 and ML = 9, then LN² = 9² + 9² = 81 + 81 = 162. Hence LN = √162 = √(81 × 2) = 9√2 cm, which matches the given hypotenuse length. The result is correct.

Why Other Options Are Wrong:
The value 18 cm would make the hypotenuse longer than 9√2 cm. Values 9√2 or 9/√2 do not match the required leg length given the standard 45°–45°–90° ratio. Only 9 cm gives the correct hypotenuse when plugged back into the Pythagorean theorem.

Common Pitfalls:
A common mistake is to think that the hypotenuse is double a leg rather than a√2 times a leg, or to mix up which side is the hypotenuse. Another frequent error is failing to simplify the expression when dividing by √2. Remember that in a 45°–45°–90° triangle, the hypotenuse is exactly √2 times any leg.

Final Answer:
The length of MN is 9 cm.