Find the area of an isosceles triangle which has:
a . Base 16 cm and equal sides 17 cm each.
b. base 24 cm and equal sides 20 cm each.

Answer

To find the area of the isosceles triangles using Heron's formula, we will follow these steps for both parts of the question.

### Part A: Base = 16 cm, Equal Sides = 17 cm

1. **Identify the sides of the triangle**:
- Let the equal sides \( A = 17 \) cm and \( B = 17 \) cm, and the base \( C = 16 \) cm.

2. **Calculate the semi-perimeter (s)**:
\[ s = \frac{A + B + C}{2} = \frac{17 + 17 + 16}{2} = \frac{50}{2} = 25 \text{ cm} \]

3. **Apply Heron's formula for area**:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ \text{Area} = \sqrt{25(25 - 17)(25 - 17)(25 - 16)} \]
\[ = \sqrt{25 \times 8 \times 8 \times 9} \]

4. **Calculate the area**:
- First, simplify under the square root:
\[ = \sqrt{25} \times \sqrt{8^2} \times \sqrt{9} \]
\[ = 5 \times 8 \times 3 = 120 \text{ cm}^2 \]

### Part B: Base = 24 cm, Equal Sides = 20 cm

1. **Identify the sides of the triangle**:
- Let the equal sides \( A = 20 \) cm and \( B = 20 \) cm, and the base \( C = 24 \) cm.

2. **Calculate the semi-perimeter (s)**:
\[ s = \frac{A + B + C}{2} = \frac{20 + 20 + 24}{2} = \frac{64}{2} = 32 \text{ cm} \]

3. **Apply Heron's formula for area**:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ \text{Area} = \sqrt{32(32 - 20)(32 - 20)(32 - 24)} \]
\[ = \sqrt{32 \times 12 \times 12 \times 8} \]

4. **Calculate the area**:
- First, simplify under the square root:
\[ = \sqrt{32} \times \sqrt{12^2} \times \sqrt{8} \]
\[ = 4\sqrt{2} \times 12 \times 2\sqrt{2} = 4 \times 12 \times 2 \times 2 = 192 \text{ cm}^2 \]

### Final Answers:
- Area of triangle in Part A: **120 cm²**
- Area of triangle in Part B: **192 cm²**