In ΔLMN, l = 77 cm, m = 17 cm and n=77 cm. Find the area of ΔLMN to the nearest square centimeter.

The triangle \( \Delta LMN \) has sides \( l = 77 \, \text{cm} \), \( m = 17 \, \text{cm} \), and \( n = 77 \, \text{cm} \). Notice that two sides are equal (\( l = n \)), indicating that the triangle is isosceles.

We can use **Heron's formula** to find the area of the triangle. Heron's formula is given by:

\[
\text{Area} = \sqrt{s(s - l)(s - m)(s - n)}
\]

where \( s \) is the semi-perimeter of the triangle:

\[
s = \frac{l + m + n}{2}
\]

### Step 1: Calculate the semi-perimeter \( s \)

\[
s = \frac{77 + 17 + 77}{2} = \frac{171}{2} = 85.5 \, \text{cm}
\]

### Step 2: Apply Heron's formula

\[
\text{Area} = \sqrt{85.5(85.5 - 77)(85.5 - 17)(85.5 - 77)}
\]

\[
\text{Area} = \sqrt{85.5 \times 8.5 \times 68.5 \times 8.5}
\]

### Step 3: Simplify

\[
\text{Area} = \sqrt{85.5 \times 8.5^2 \times 68.5}
\]

Calculating this value:

\[
\text{Area} \approx 660 \, \text{square centimeters}
\]

Thus, the area of \( \Delta LMN \) is approximately **660 square centimeters**.

Would you like more details or have any questions?

### Related Questions:
1. How is Heron's formula derived?
2. What properties distinguish an isosceles triangle from other triangles?
3. How do you find the height of an isosceles triangle given its sides?
4. How does Heron's formula change for equilateral triangles?
5. Can Heron's formula be used for right triangles?

### Tip
If two sides of a triangle are equal, use symmetry properties to simplify calculations such as height or area estimation.

In ΔLMN, l = 77 cm, m = 17 cm and n = 77 cm. Find the area of ΔLMN to the nearest square centimeter.

Learn how to calculate the area of an isosceles triangle with sides 77 cm, 17 cm, and 77 cm using Heron's formula. This step-by-step guide explains the process of finding the semi-perimeter and applying Heron's theorem to solve for the area, making it suitable for high school students in Grades 9-12.

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