1. **Problem Statement:** Given a rectangle DEFG with side GF = 11 and side GH = 14, find the lengths GE, DF, HF, and DG. 2. **Recall Properties of Rectangles:** - Opposite sides are equal in length. - Diagonals are equal in length. - Diagonals bisect each other. 3. **Assign Known Values:** - GF = 11 (given) - GH = 14 (given) 4. **Find GE and DF:** Since GE and DF are opposite sides to GF and GH respectively, we have: $$GE = GF = 11$$ $$DF = GH = 14$$ 5. **Find the length of diagonal DG (or EF):** Using the Pythagorean theorem for right triangle formed by sides GF and GH: $$DG = \sqrt{GF^2 + GH^2} = \sqrt{11^2 + 14^2} = \sqrt{121 + 196} = \sqrt{317}$$ 6. **Find HF (half of diagonal DG):** Since diagonals bisect each other at H: $$HF = \frac{DG}{2} = \frac{\sqrt{317}}{2}$$ **Final answers:** $$GE = 11$$ $$DF = 14$$ $$DG = \sqrt{317}$$ $$HF = \frac{\sqrt{317}}{2}$$