`DeltaLMN `is right angled at M. If `M /_N = 45^@`, what is the length of MN (in cm), if `NL =9sqrt(2) cm`?

Options

A. `9sqrt(2)`
B. `9/(sqrt2)`
C. `18`
D. `9`

Answer

To solve the problem, we need to analyze the right-angled triangle \( \Delta LMN \) where \( M \) is the right angle and \( \angle N = 45^\circ \). Given that \( NL = 9\sqrt{2} \) cm, we can find the length of \( MN \).

### Step-by-Step Solution:

1. **Identify the Triangle Properties**:
Since \( \Delta LMN \) is a right-angled triangle at \( M \) and \( \angle N = 45^\circ \), it follows that \( \angle L \) must also be \( 45^\circ \) because the angles in a triangle sum up to \( 180^\circ \). Therefore, \( \Delta LMN \) is an isosceles right triangle.

**Hint**: In a right-angled triangle with one angle of \( 45^\circ \), the other non-right angle is also \( 45^\circ \).

2. **Use the Properties of Isosceles Right Triangles**:
In an isosceles right triangle, the lengths of the two legs are equal. Let \( MN = x \) and \( LM = x \).

**Hint**: In an isosceles right triangle, the legs opposite the \( 45^\circ \) angles are equal.

3. **Apply the Pythagorean Theorem**:
The hypotenuse \( NL \) can be expressed using the Pythagorean theorem:
\[ NL^2 = MN^2 + LM^2 \]
Substituting \( MN = x \) and \( LM = x \):
\[ NL^2 = x^2 + x^2 = 2x^2 \]
Given \( NL = 9\sqrt{2} \):
\[ (9\sqrt{2})^2 = 2x^2 \]
\[ 81 \cdot 2 = 2x^2 \]
\[ 162 = 2x^2 \]
Dividing both sides by 2:
\[ x^2 = 81 \]
Taking the square root of both sides:
\[ x = 9 \]

**Hint**: Remember to square both sides when using the Pythagorean theorem and simplify correctly.

4. **Conclusion**:
Therefore, the length of \( MN \) is \( 9 \) cm.

**Final Answer**: \( MN = 9 \) cm.