1. **Problem Statement:** We need to find the length of segment $LK$ in a diamond-shaped figure with points $L$, $K$, $N$, and $M$. Given that $KL = x$ and $KN = 2x - 3$, and there are right angles at $L$ and $N$, indicating perpendicular lines. 2. **Understanding the Figure:** Since $L$ and $N$ have right angles, and the figure is a diamond (a rhombus), all sides are equal in length. Therefore, $LK = KN = LM = MN$. 3. **Set up the equation:** Since $LK = x$ and $KN = 2x - 3$, and these are equal sides, we have: $$x = 2x - 3$$ 4. **Solve for $x$:** $$x = 2x - 3$$ $$x - 2x = -3$$ $$-x = -3$$ $$x = 3$$ 5. **Find $LK$:** Since $LK = x$, then: $$LK = 3$$ **Final answer:** $LK = 3$