Question

the piremeter of the parallelogram ABCD is equal to 10cm. Find the length of the diaganol line BD if you know that the permimeter of the triangle ABD is 8cm

Answer 1

To find the length of diagonal \( BD \) in parallelogram \( ABCD \), given the perimeter of the parallelogram is \( 10 \, \text{cm} \) and the perimeter of triangle \( ABD \) is \( 8 \, \text{cm} \), we can use the following steps:

1. **Properties of Parallelograms**:
- The opposite sides of a parallelogram are equal in length.
- Thus, if we denote the lengths of sides \( AB \) and \( AD \) by \( a \) and \( b \), respectively, the perimeter of the parallelogram is given by:
\[
2(a + b) = 10 \implies a + b = 5 \quad \text{(1)}
\]

2. **Perimeter of Triangle \( ABD \)**:
- The perimeter of triangle \( ABD \) is given by:
\[
AB + AD + BD = 8 \implies a + b + BD = 8 \quad \text{(2)}
\]

3. **Setting Up the Equations**:
- From equation (1): \( a + b = 5 \).
- Substituting \( a + b \) from (1) into (2) gives:
\[
5 + BD = 8
\]

4. **Solving for \( BD \)**:
- Rearranging the equation:
\[
BD = 8 - 5 = 3
\]

Therefore, the length of diagonal \( BD \) is \( \boxed{3 \, \text{cm}} \).