1. **State the problem:** We are given a triangle WUY with points V and X as midpoints of segments UW and WY respectively. We know: - $UY = t - 42$ - $VX = t - 51$ We need to find the value of $VX$. 2. **Understand the midpoint theorem:** The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. 3. **Apply the midpoint theorem:** Since $V$ and $X$ are midpoints of $UW$ and $WY$, segment $VX$ is parallel to $UY$ and: $$VX = \frac{1}{2} UY$$ 4. **Set up the equation:** $$t - 51 = \frac{1}{2} (t - 42)$$ 5. **Solve for $t$:** Multiply both sides by 2: $$2(t - 51) = t - 42$$ $$2t - 102 = t - 42$$ Subtract $t$ from both sides: $$t - 102 = -42$$ Add 102 to both sides: $$t = 60$$ 6. **Find $VX$ by substituting $t=60$:** $$VX = t - 51 = 60 - 51 = 9$$ **Final answer:** $$VX = 9$$