The area of an isosceles triangle is `9 cm^(2)`. If the equal sides are 6 cm in length, the angle between them is

Options

A. `60^(@)`
B. `30^(@)`
C. `90^(@)`
D. `45^(@)`

Answer

To find the angle between the equal sides of the isosceles triangle given its area and the lengths of the sides, we can use the formula for the area of a triangle in terms of two sides and the included angle.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Area (A) = 9 cm²
- Length of equal sides (a) = 6 cm

2. **Use the Area Formula:**
The area of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times a \times b \times \sin(C) \]
where \(a\) and \(b\) are the lengths of the two sides, and \(C\) is the included angle between them. In this case, since the triangle is isosceles, we have:
\[ A = \frac{1}{2} \times 6 \times 6 \times \sin(C) \]

3. **Substitute the Values into the Area Formula:**
\[ 9 = \frac{1}{2} \times 6 \times 6 \times \sin(C) \]
Simplifying this, we get:
\[ 9 = 18 \sin(C) \]

4. **Solve for \(\sin(C)\):**
Divide both sides by 18:
\[ \sin(C) = \frac{9}{18} = \frac{1}{2} \]

5. **Find the Angle \(C\):**
The angle \(C\) whose sine is \(\frac{1}{2}\) is:
\[ C = 30^\circ \]

### Final Answer:
The angle between the equal sides of the isosceles triangle is \(30^\circ\).