1. **State the problem:** We need to find the value of $x$ such that the expressions for the top-left and top-right are equal: $$\sqrt{100^2 - x^2} = \sqrt{89.4^2 - (x-20)^2}$$ 2. **Square both sides** to eliminate the square roots: $$100^2 - x^2 = 89.4^2 - (x-20)^2$$ 3. **Expand the squared term on the right:** $$(x-20)^2 = x^2 - 2 \cdot 20 \cdot x + 20^2 = x^2 - 40x + 400$$ 4. **Substitute back and simplify:** $$10000 - x^2 = 7992.36 - (x^2 - 40x + 400)$$ $$10000 - x^2 = 7992.36 - x^2 + 40x - 400$$ 5. **Add $x^2$ to both sides to cancel:** $$10000 = 7992.36 + 40x - 400$$ 6. **Simplify constants:** $$10000 = 7592.36 + 40x$$ 7. **Isolate $x$:** $$10000 - 7592.36 = 40x$$ $$2407.64 = 40x$$ 8. **Divide both sides by 40:** $$x = \frac{2407.64}{40}$$ $$x = \cancel{\frac{2407.64}{\cancel{40}}} = 60.191$$ **Final answer:** $$x \approx 60.19$$