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We are given a right triangle with an area of 54 cm², and one of its legs (AB) is 12 cm long. We are asked to find the perimeter of the triangle.

### Step 1: Use the area formula for a right triangle
The area \( A \) of a right triangle is given by:

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

Here, we know:
- \( A = 54 \, \text{cm}^2 \)
- One leg (AB) = 12 cm (which we can take as the height)

Let the other leg (BC) be \( b \). The formula becomes:

\[
54 = \frac{1}{2} \times 12 \times b
\]

Simplifying:

\[
54 = 6 \times b
\]

\[
b = \frac{54}{6} = 9 \, \text{cm}
\]

Thus, the other leg \( BC = 9 \, \text{cm} \).

### Step 2: Use the Pythagorean theorem to find the hypotenuse
For a right triangle, the Pythagorean theorem states:

\[
AC^2 = AB^2 + BC^2
\]

Substitute the known values:

\[
AC^2 = 12^2 + 9^2 = 144 + 81 = 225
\]

\[
AC = \sqrt{225} = 15 \, \text{cm}
\]

### Step 3: Find the perimeter
The perimeter \( P \) of the triangle is the sum of all its sides:

\[
P = AB + BC + AC = 12 + 9 + 15 = 36 \, \text{cm}
\]

Thus, the perimeter of the triangle is **36 cm**.

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:
1. How is the area formula for a right triangle derived?
2. Can the Pythagorean theorem be used for triangles other than right triangles?
3. How would the perimeter change if the area remained the same but one leg was shorter?
4. What is the relationship between the area and the hypotenuse length of a right triangle?
5. How would you find the area if the hypotenuse and one leg were given instead?

**Tip:** When solving for triangle dimensions, double-check if it’s a right triangle, since it allows the use of the Pythagorean theorem for quick calculations.

Area of a right triangle is 54 cm². If one of its legs is 12 cm long, its perimeter is?

Learn how to calculate the perimeter of a right triangle with a given area of 54 cm² and one leg measuring 12 cm. This problem uses the area formula for triangles and the Pythagorean theorem, suitable for Grades 6-8.

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