The sides of triangle ABC are 5,5,6 and the side of triangle PQR are 5,5,8. Find the correct relation between the areas of two triangles.

Options

A. `triangleABC gt trianglePQR`
B. `triangleABC lt trianglePQR`
C. `triangleABC = trianglePQR`
D. can't be determined

Answer

To find the correct relationship between the areas of triangles ABC and PQR, we can use the formula for the area of an isosceles triangle. The formula for the area \( A \) of an isosceles triangle with two equal sides \( a \) and base \( b \) is given by:

\[ A = \frac{1}{2} \times b \times \sqrt{a^2 - \left(\frac{b}{2}\right)^2} \]

### Step 1: Calculate the area of triangle ABC
1. **Identify the sides**: For triangle ABC, the sides are \( a = 5 \), \( a = 5 \), and \( b = 6 \).
2. **Substitute into the area formula**:
\[ A_{ABC} = \frac{1}{2} \times 6 \times \sqrt{5^2 - \left(\frac{6}{2}\right)^2} \]
3. **Calculate \( \left(\frac{6}{2}\right)^2 \)**:
\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \]
4. **Calculate \( 5^2 - 9 \)**:
\[ 5^2 = 25 \quad \Rightarrow \quad 25 - 9 = 16 \]
5. **Calculate the square root**:
\[ \sqrt{16} = 4 \]
6. **Calculate the area**:
\[ A_{ABC} = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2 \]

### Step 2: Calculate the area of triangle PQR
1. **Identify the sides**: For triangle PQR, the sides are \( a = 5 \), \( a = 5 \), and \( b = 8 \).
2. **Substitute into the area formula**:
\[ A_{PQR} = \frac{1}{2} \times 8 \times \sqrt{5^2 - \left(\frac{8}{2}\right)^2} \]
3. **Calculate \( \left(\frac{8}{2}\right)^2 \)**:
\[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
4. **Calculate \( 5^2 - 16 \)**:
\[ 5^2 = 25 \quad \Rightarrow \quad 25 - 16 = 9 \]
5. **Calculate the square root**:
\[ \sqrt{9} = 3 \]
6. **Calculate the area**:
\[ A_{PQR} = \frac{1}{2} \times 8 \times 3 = 12 \text{ cm}^2 \]

### Step 3: Compare the areas
- The area of triangle ABC is \( 12 \text{ cm}^2 \).
- The area of triangle PQR is also \( 12 \text{ cm}^2 \).

### Conclusion
The areas of both triangles are equal:
\[ A_{ABC} = A_{PQR} \]