1. **Problem statement:** Given a right triangle أ د ج with a right angle at د and side lengths ٤ and ٤, find the length of the hypotenuse أ ج and verify the length of segment ه ج using the Pythagorean theorem. 2. **Formula used:** For a right triangle with legs $a$ and $b$, the hypotenuse $c$ is given by $$c = \sqrt{a^2 + b^2}$$ 3. **Calculate length أ ج:** Since أ د = ٤ and د ج = ٤, then $$أ ج = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$$ 4. **Square of أ ج:** $$(أ ج)^2 = 32$$ 5. **Calculate length ه ج:** Given ه ج is the hypotenuse of a right triangle with legs ٧ and ٤٩, apply Pythagoras: $$ه ج = \sqrt{7^2 + 49} = \sqrt{49 + 49} = \sqrt{98} \approx 9.9$$ 6. **Conclusion:** The length أ ج is $4\sqrt{2}$, approximately 5.66, and the length ه ج is approximately 9.9, confirming the calculations.