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We are given a triangle \( \Delta LMN \) with the following information:

- \( MP = 9 \) units
- \( LP = 14 \) units
- Area of \( \Delta LMN \) = 243 square units

We are asked to find the length of \( MN \).

### Step-by-Step Solution:

The area of triangle \( \Delta LMN \) is given by:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

In this case:
- The base \( MN \) is what we are trying to find.
- The height is \( MP = 9 \) units.
- The area is \( 243 \) square units.

Thus, the formula becomes:

\[
243 = \frac{1}{2} \times MN \times 9
\]

Solving for \( MN \):

\[
243 = \frac{9 \times MN}{2}
\]

Multiply both sides by 2 to eliminate the fraction:

\[
486 = 9 \times MN
\]

Now divide both sides by 9:

\[
MN = \frac{486}{9} = 54 \, \text{units}
\]

Thus, the length of \( MN \) is **54 units**.

Would you like further explanation or have any questions?

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Here are 5 related questions to expand on the topic:

1. How do you calculate the area of a triangle given all three sides (Heron's formula)?
2. What happens if we change the height while keeping the base and area the same?
3. How can we find the angles of triangle \( \Delta LMN \) using the information given?
4. How would you apply trigonometry to solve for unknown angles in \( \Delta LMN \)?
5. Can we use Pythagoras' theorem in any part of this triangle, given it has a right angle at \( P \)?

**Tip:** Always remember that the area formula for a triangle can involve any side as the base, as long as the corresponding height is perpendicular to that side.

In this diagram, MP = 9 units and LP = 14 units. The area of triangle LMN is 243 square units. What is the length of MN?

Learn how to find the length of side MN in triangle LMN when given the height MP = 9 units, base LP = 14 units, and the area as 243 square units. This problem involves applying the area of a triangle formula and basic algebra to solve for MN. Suitable for students in Grades 8-10.

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