In right triangle LMN, right-angled at M, angle N is 45°. If the hypotenuse LN is 9√2 cm, what is the length of side MN (in cm)?

Introduction / Context:
Right triangles with angles 45°, 45°, and 90° are very common in aptitude and geometry questions. Such triangles are isosceles right triangles, meaning the two legs are equal in length and the hypotenuse is longer by a fixed factor. Recognizing this pattern allows you to quickly calculate side lengths without using trigonometric functions. This question checks whether you can use the standard side ratio for a 45°–45°–90° triangle correctly.

Given Data / Assumptions:

• Triangle LMN is right angled at M.
• Angle N is 45°, so angle L is also 45°.
• The hypotenuse LN is given as 9√2 cm.
• We need to find the length of side MN.

Concept / Approach:
In a 45°–45°–90° right triangle, the two legs are equal in length. If each leg has length a, then the hypotenuse has length a√2. Therefore, the relationship is:
hypotenuse = leg * √2 From this we can deduce:
leg = hypotenuse / √2 Since triangle LMN is right angled at M and angle N is 45°, sides LM and MN are the equal legs, and LN is the hypotenuse.

Step-by-Step Solution:
Step 1: Recognize that angles at L and N are both 45°, making triangle LMN an isosceles right triangle. Step 2: Let each leg (including MN) have length a. Step 3: Then the hypotenuse LN must be a√2. Step 4: We are told LN = 9√2 cm. Step 5: Equate: a√2 = 9√2. Step 6: Divide both sides by √2 to obtain a = 9. Step 7: Since MN is one of the legs, MN = 9 cm.

Verification / Alternative check:
We can check using the Pythagorean theorem. If each leg is 9 cm, then:
LN^2 = LM^2 + MN^2 = 9^2 + 9^2 = 81 + 81 = 162. Then LN = sqrt(162) = sqrt(81 * 2) = 9√2 cm, which matches the given hypotenuse. So the value of 9 cm for MN is consistent with the right triangle condition.

Why Other Options Are Wrong:

• 9√2: This is the hypotenuse length, not the leg length.
• 9/√2: This would be smaller than the correct leg, and does not match the standard 45°–45°–90° ratios for the given hypotenuse.
• 18: This would make the hypotenuse even longer than 18√2, which conflicts with the given 9√2.
• 6√2: This would not produce the given hypotenuse when used in the Pythagorean theorem.

Common Pitfalls:
A common error is to reverse the relationship and assume the hypotenuse is leg / √2. Others forget that in a 45°–45°–90° triangle, the legs must be equal. Remembering the pattern "leg, leg, leg√2" helps solve such questions quickly and accurately.

Final Answer:
Thus, the length of side MN is 9 cm.