In a parallelogram the diagonals are equal in length

Answer

To determine whether the statement "In a parallelogram, the diagonals are equal in length" is true or false, we can analyze the properties of a parallelogram and its diagonals mathematically.

### Step-by-Step Solution:

1. **Understanding the Parallelogram**:
A parallelogram is a quadrilateral with opposite sides that are equal and parallel. Let’s denote the vertices of the parallelogram as A, B, C, and D, where AB is parallel to CD and AD is parallel to BC.

2. **Defining the Vectors**:
Let the vectors representing the sides of the parallelogram be:
- \( \vec{A} \) for side AB
- \( \vec{B} \) for side AD

3. **Finding the Diagonals**:
The diagonals of the parallelogram can be represented as:
- Diagonal \( D_1 \) from A to C: \( \vec{D_1} = \vec{A} + \vec{B} \)
- Diagonal \( D_2 \) from B to D: \( \vec{D_2} = \vec{A} - \vec{B} \)

4. **Calculating the Lengths of the Diagonals**:
To find the lengths of the diagonals, we calculate the magnitudes:
- Length of \( D_1 \):
\[ |D_1| = |\vec{A} + \vec{B}| \]
- Length of \( D_2 \):
\[ |D_2| = |\vec{A} - \vec{B}| \]

5. **Using the Formula for Magnitude**:
The magnitude of a vector can be calculated using the formula:
\[ | \vec{X} | = \sqrt{X_x^2 + X_y^2} \]
Thus, we can express the lengths of the diagonals as:
\[ |D_1|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos(\theta) \]
\[ |D_2|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2 |\vec{A}| |\vec{B}| \cos(\theta) \]
where \( \theta \) is the angle between vectors \( \vec{A} \) and \( \vec{B} \).

6. **Comparing the Lengths**:
For the diagonals to be equal, we need:
\[ |D_1|^2 = |D_2|^2 \]
This leads to:
\[ |\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos(\theta) = |\vec{A}|^2 + |\vec{B}|^2 - 2 |\vec{A}| |\vec{B}| \cos(\theta) \]
Simplifying this gives:
\[ 4 |\vec{A}| |\vec{B}| \cos(\theta) = 0 \]
This implies that \( \cos(\theta) = 0 \), meaning \( \theta = 90^\circ \).

7. **Conclusion**:
The only case when the diagonals of a parallelogram are equal is when the parallelogram is a rectangle (where the angles are right angles). Therefore, in general, the statement "In a parallelogram, the diagonals are equal in length" is **false**.

### Final Answer:
The statement is **false**.