To solve the problem of determining what the line segments joining the midpoints of the adjacent sides of a quadrilateral form, we can follow these steps:
### Step-by-Step Solution:
1. **Draw a Quadrilateral**:
Start by drawing a quadrilateral ABCD. Label the vertices as A, B, C, and D.
**Hint**: Make sure to label the vertices clearly for reference in the next steps.
2. **Identify Midpoints**:
Find the midpoints of each side of the quadrilateral. Let P be the midpoint of side AB, Q be the midpoint of side BC, R be the midpoint of side CD, and S be the midpoint of side DA.
**Hint**: Use the midpoint formula if necessary, which is \((x_1 + x_2)/2, (y_1 + y_2)/2\).
3. **Draw Line Segments**:
Draw line segments joining the midpoints: PQ, QR, RS, and SP.
**Hint**: Ensure that you are connecting the midpoints in the correct order to form a closed shape.
4. **Analyze Triangles**:
Look at triangles formed by the vertices of the quadrilateral and the midpoints. For example, in triangle ABC, P and Q are midpoints of sides AB and BC, respectively.
**Hint**: Remember that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
5. **Establish Parallel Lines**:
From the previous step, since PQ is parallel to AC (the third side of triangle ABC) and RS is parallel to AC (in triangle ACD), we conclude that PQ is parallel to RS.
**Hint**: Use the properties of parallel lines and transversals to reinforce your conclusions.
6. **Repeat for Other Triangles**:
Now consider triangles ABD and BCD. You will find that PS is parallel to QR by similar reasoning.
**Hint**: Always check the relationships in both triangles to ensure consistency.
7. **Conclude the Shape**:
Since both pairs of opposite sides (PQ and RS, PS and QR) are parallel, we can conclude that the quadrilateral formed by the midpoints P, Q, R, and S is a parallelogram.
**Hint**: Remember that a quadrilateral is a parallelogram if both pairs of opposite sides are parallel.
### Final Conclusion:
The line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.
