The base of an isosceles triangle is 12 cm and its perimeter is 32 cm. Find the area of the triangle.

Answer

To find the area of the isosceles triangle with a base of 12 cm and a perimeter of 32 cm, we can follow these steps:

### Step 1: Understand the triangle's properties
We know that in an isosceles triangle, two sides are equal. Let's denote the equal sides as \( x \) and the base as \( BC = 12 \) cm.

### Step 2: Set up the perimeter equation
The perimeter of the triangle is the sum of all its sides:
\[ \text{Perimeter} = x + x + 12 = 2x + 12 \]
Given that the perimeter is 32 cm, we can set up the equation:
\[ 2x + 12 = 32 \]

### Step 3: Solve for \( x \)
Subtract 12 from both sides:
\[ 2x = 32 - 12 \]
\[ 2x = 20 \]
Now, divide by 2:
\[ x = 10 \text{ cm} \]
So, the lengths of the two equal sides are each 10 cm.

### Step 4: Draw the altitude
Next, we will draw the altitude from the vertex opposite the base to the midpoint of the base. Let's denote the midpoint of the base \( BC \) as point \( D \). Since \( BC = 12 \) cm, \( BD = DC = 6 \) cm.

### Step 5: Apply the Pythagorean theorem
In triangle \( ABD \) (where \( A \) is the vertex opposite the base), we can use the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ 10^2 = h^2 + 6^2 \]
This simplifies to:
\[ 100 = h^2 + 36 \]
Now, isolate \( h^2 \):
\[ h^2 = 100 - 36 \]
\[ h^2 = 64 \]
Taking the square root gives us:
\[ h = 8 \text{ cm} \]

### Step 6: Calculate the area of the triangle
Now, we can find the area using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 12 \times 8 \]
Calculating this gives:
\[ \text{Area} = \frac{1}{2} \times 96 = 48 \text{ cm}^2 \]

### Final Answer:
The area of the triangle is \( 48 \text{ cm}^2 \).

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