The area of an isosceles triangle ABC with `AB = AC` and altitude `AD = 3 cm` is `12 sq cm`. What is its perimeter?

Options

A. 18 cm
B. 6 cm
C. 14 cm
D. 12 cm

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Understand the given information
We have an isosceles triangle ABC where:
- \( AB = AC \)
- The altitude \( AD = 3 \, \text{cm} \)
- The area of triangle ABC is \( 12 \, \text{cm}^2 \)

### Step 2: Use the area formula for triangles
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, the height (altitude) is \( AD = 3 \, \text{cm} \) and the area is \( 12 \, \text{cm}^2 \). We can set up the equation:
\[ 12 = \frac{1}{2} \times \text{base} \times 3 \]

### Step 3: Solve for the base
Rearranging the equation to find the base:
\[ 12 = \frac{3}{2} \times \text{base} \]
Multiplying both sides by \( \frac{2}{3} \):
\[ \text{base} = \frac{12 \times 2}{3} = \frac{24}{3} = 8 \, \text{cm} \]

### Step 4: Identify the lengths of the sides
Since ABC is an isosceles triangle, we denote:
- \( AB = AC = x \)
- The base \( BC = 8 \, \text{cm} \)

### Step 5: Apply the Pythagorean theorem
In triangle \( ABD \) (where \( D \) is the foot of the altitude from \( A \) to \( BC \)):
- \( BD = \frac{BC}{2} = \frac{8}{2} = 4 \, \text{cm} \)
- \( AD = 3 \, \text{cm} \)

Using the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ x^2 = 3^2 + 4^2 \]
\[ x^2 = 9 + 16 = 25 \]
Taking the square root:
\[ x = \sqrt{25} = 5 \, \text{cm} \]
Thus, \( AB = AC = 5 \, \text{cm} \).

### Step 6: Calculate the perimeter
The perimeter \( P \) of triangle ABC is given by:
\[ P = AB + AC + BC = 5 + 5 + 8 = 18 \, \text{cm} \]

### Final Answer
The perimeter of triangle ABC is \( 18 \, \text{cm} \).

---