Find the area of an isosceles triangle with perimeter 36 cm and base 16 cm.

Answer

To find the area of an isosceles triangle with a perimeter of 36 cm and a base of 16 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle Properties**:
- We have an isosceles triangle with a base \( BC = 16 \) cm.
- Let the equal sides be \( AB = AC = A \).

2. **Set Up the Perimeter Equation**:
- The perimeter of the triangle is given by the formula:
\[ \text{Perimeter} = AB + AC + BC = A + A + 16 = 2A + 16 \]
- According to the problem, the perimeter is 36 cm:
\[ 2A + 16 = 36 \]

3. **Solve for A**:
- Rearranging the equation:
\[ 2A = 36 - 16 \]
\[ 2A = 20 \]
\[ A = \frac{20}{2} = 10 \text{ cm} \]

4. **Determine the Height of the Triangle**:
- The height \( h \) of the triangle can be found by drawing a perpendicular from point A to the base BC, meeting at point D. This divides the base into two equal segments:
\[ BD = DC = \frac{BC}{2} = \frac{16}{2} = 8 \text{ cm} \]
- Now, we have a right triangle ABD where:
- \( AB = 10 \) cm (the equal side),
- \( BD = 8 \) cm (half of the base),
- \( AD = h \) (the height).
- Using the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \]
\[ 10^2 = h^2 + 8^2 \]
\[ 100 = h^2 + 64 \]
\[ h^2 = 100 - 64 = 36 \]
\[ h = \sqrt{36} = 6 \text{ cm} \]

5. **Calculate the Area of the Triangle**:
- The area \( A \) of the triangle is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Substituting the known values:
\[ \text{Area} = \frac{1}{2} \times 16 \times 6 \]
\[ \text{Area} = 8 \times 6 = 48 \text{ cm}^2 \]

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

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