1. **State the problem:** We have three right triangles △ABE, △BEC, and △ADC with given side lengths and need to find the value of $x$. 2. **Identify known sides and right angles:** - In △ABE, right angle at E, sides $AE=7$, $BE=11$. - In △BEC, right angle at E, side $CE=14$. - In △ADC, right angle at D, side $AD=13$, side $DC=x$. 3. **Use the Pythagorean theorem:** For a right triangle with legs $a$, $b$ and hypotenuse $c$, the relation is $$a^2 + b^2 = c^2$$ 4. **Find $AB$ in △ABE:** $$AB^2 = AE^2 + BE^2 = 7^2 + 11^2 = 49 + 121 = 170$$ $$AB = \sqrt{170}$$ 5. **Find $BC$ in △BEC:** Since $E$ is right angle, $BC^2 = BE^2 + CE^2 = 11^2 + 14^2 = 121 + 196 = 317$ $$BC = \sqrt{317}$$ 6. **Find $AC$ using $AB$ and $BC$:** Points $A$, $B$, and $C$ are connected, so $$AC = AB + BC = \sqrt{170} + \sqrt{317}$$ 7. **Use △ADC with right angle at D:** $$AC^2 = AD^2 + DC^2$$ Substitute known values: $$\left(\sqrt{170} + \sqrt{317}\right)^2 = 13^2 + x^2$$ 8. **Expand the left side:** $$\left(\sqrt{170} + \sqrt{317}\right)^2 = 170 + 2\sqrt{170 \times 317} + 317 = 487 + 2\sqrt{53890}$$ 9. **Calculate $\sqrt{53890}$:** $$\sqrt{53890} \approx 232.19$$ So, $$487 + 2 \times 232.19 = 487 + 464.38 = 951.38$$ 10. **Set up equation for $x^2$:** $$951.38 = 169 + x^2$$ 11. **Solve for $x^2$:** $$x^2 = 951.38 - 169 = 782.38$$ 12. **Find $x$:** $$x = \sqrt{782.38} \approx 27.97$$ **Final answer:** $$\boxed{27.97}$$