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

Question

Accepted Answer

To determine the area of the isosceles triangle with a base of 12 cm and a perimeter of 32 cm, we first need to find the lengths of the equal sides.

### Step-by-Step Explanation:

1. **Calculate the Length of the Equal Sides:**
   - The perimeter of the triangle is given as 32 cm.
   - Let the length of each of the equal sides be $ x $.
   - The perimeter of the triangle is the sum of all its sides: $ x + x + 12 = 32 $.
   - Simplifying, we get: $ 2x + 12 = 32 $.
   - Solving for $ x $, we subtract 12 from both sides: $ 2x = 20 $.
   - Divide by 2: $ x = 10 $.
   - Therefore, each of the equal sides is 10 cm long.

2. **Calculate the Height of the Triangle:**
   - To find the area, we need the height of the triangle. Since the triangle is isosceles, the height will bisect the base into two equal parts of 6 cm each.
   - We can use the Pythagorean theorem to find the height ($ h $) of the triangle:
     $$
     h^2 + 6^2 = 10^2
     $$
     $$
     h^2 + 36 = 100
     $$
     $$
     h^2 = 64
     $$
     $$
     h = 8
     $$

3. **Calculate the Area of the Triangle:**
   - The area ($ A $) of a triangle is given by the formula:
     $$
     A = \frac{1}{2} \times \text{base} \times \text{height}
     $$
   - Substituting the known values:
     $$
     A = \frac{1}{2} \times 12 \times 8 = 48 \text{ cm}^2
     $$

### Justification for Option 2:

- The calculated area of the isosceles triangle is $ 48 \text{ cm}^2 $, which matches Option 2. Therefore, Option 2 is the correct answer.

### Explanation for Incorrect Options:

- **Option 1: $42 \text{ cm}^2$** - This value is incorrect because the calculated area using the correct dimensions and formula does not yield 42 cm².
- **Option 3: $54 \text{ cm}^2$** - This value is incorrect as it does not match the area calculated using the given dimensions.
- **Option 4: $60 \text{ cm}^2$** - This value is incorrect because the area calculation based on the given side lengths does not result in 60 cm².

In conclusion, the correct area of the isosceles triangle, based on the given dimensions and calculations, is $ 48 \text{ cm}^2 $, which corresponds to Option 2.