To construct a quadrilateral ABCD, where AB = 4 cm, BC = 6 cm, CD = DA = 5.2 cm, and AC = 8 cm, follow the steps below:

1. Draw a line segment AB of length 4 cm.
2. From point A, draw an angle of 120 degrees using a protractor.
3. From point B, draw an angle of 60 degrees using a protractor.
4. Let the lines from points A and B intersect at point O.
5. From point O, draw a line segment OC of length 8 cm.
6. From point C, draw an angle of 60 degrees using a protractor.
7. From point O, draw a line segment OD of length 5.2 cm.
8. From point D, draw an angle of 120 degrees using a protractor.
9. Let the lines from points C and D intersect at point P.
10. Connect points A, B, C, and D to form the quadrilateral ABCD.

Explanation:

- The given information states that AB = 4 cm, BC = 6 cm, CD = DA = 5.2 cm, and AC = 8 cm. These lengths will be used to construct the quadrilateral ABCD.
- Steps 1 and 2 involve drawing a line segment AB and constructing angles at points A and B using a protractor.
- Step 4 involves determining the point of intersection, which we will call point O. This point will be used as a reference for constructing the quadrilateral.
- In Step 5, a line segment OC of length 8 cm is drawn from point O. This represents the side AC of the quadrilateral.
- In Step 6, an angle of 60 degrees is constructed at point C using a protractor.
- In Step 7, a line segment OD of length 5.2 cm is drawn from point O. This represents the side AD or DC of the quadrilateral.
- In Step 8, an angle of 120 degrees is constructed at point D using a protractor.
- In Step 9, the lines from points C and D intersect at point P, which is the final point needed to construct the quadrilateral.
- Finally, in Step 10, points A, B, C, and D are connected to form the quadrilateral ABCD.

By following these steps and using the given measurements, you can construct the quadrilateral ABCD.