1. **Problem Statement:** We have a geometric figure with segments labeled 9, 5, 7, 20, and an unknown length $x$. We need to find the value of $x$. 2. **Understanding the figure:** The figure consists of a vertical segment of length 9 meeting a horizontal segment of length 5 at a right angle, forming a corner. From the bottom-left corner, a diagonal segment labeled $x$ rises to the top-middle point. From this top-middle point, a slanted segment labeled 7 extends to the right endpoint, perpendicular to the previous segment. The long bottom diagonal from the bottom-left corner to the right endpoint is labeled 20. 3. **Approach:** We can use the Pythagorean theorem and properties of right triangles to find $x$. 4. **Step 1:** Consider the right triangle formed by the vertical segment 9 and horizontal segment 5. The hypotenuse of this triangle is $x$. Using the Pythagorean theorem: $$x^2 = 9^2 + 5^2$$ $$x^2 = 81 + 25$$ $$x^2 = 106$$ 5. **Step 2:** Calculate $x$: $$x = \sqrt{106}$$ 6. **Step 3:** Approximate the value: $$x \approx 10.2956$$ 7. **Final answer:** The length of $x$ is approximately $10.3$ units.