1. **State the problem:** We have two triangles with vertical segments inside them. We need to find the value of $x$ and then calculate the length of segment $VT$ in the second triangle. 2. **Analyze the first triangle EFH:** - Given sides: $EH = 9$, $HF = 9$, $EG = x + 5$, $GH = 6$, and $FD = 12$. - Since $HG$ is vertical inside the triangle, it suggests right triangles or similar triangles. 3. **Analyze the second triangle STR:** - Given sides: $SR = 14$, $ST = 3x - 3$, $UT = x + 2$, $UV = 6$. - $VU$ is vertical inside the triangle. 4. **Use similarity of triangles:** - The vertical segments inside the triangles create smaller right triangles similar to the larger ones. - For triangle EFH, the smaller triangle formed by $EG$ and $GH$ is similar to the larger triangle. 5. **Set up proportion for triangle EFH:** Since $HG$ is vertical, triangles $EGH$ and $EFH$ are similar. The ratio of corresponding sides is: $$\frac{EG}{EH} = \frac{GH}{HF}$$ Substitute values: $$\frac{x+5}{9} = \frac{6}{9}$$ 6. **Solve for $x$:** $$\frac{x+5}{9} = \frac{6}{9} \implies x + 5 = 6 \implies x = 1$$ 7. **Calculate length of $VT$ in triangle STR:** - Substitute $x=1$ into $ST$ and $UT$: $$ST = 3(1) - 3 = 0$$ $$UT = 1 + 2 = 3$$ - Since $ST=0$, this suggests a degenerate triangle or a need to re-examine the problem. 8. **Re-examine the problem:** - Possibly $ST$ should not be zero; check if $x=1$ is valid. - Alternatively, use similarity in triangle STR: Triangles $SUV$ and $STR$ are similar with vertical segment $VU$. Set up proportion: $$\frac{UV}{ST} = \frac{UT}{SR}$$ Substitute known values: $$\frac{6}{3x - 3} = \frac{x + 2}{14}$$ 9. **Solve for $x$ using this proportion:** Cross multiply: $$6 \times 14 = (3x - 3)(x + 2)$$ $$84 = (3x - 3)(x + 2)$$ Expand right side: $$84 = 3x^2 + 6x - 3x - 6$$ $$84 = 3x^2 + 3x - 6$$ Bring all terms to one side: $$3x^2 + 3x - 6 - 84 = 0$$ $$3x^2 + 3x - 90 = 0$$ Divide entire equation by 3: $$x^2 + x - 30 = 0$$ 10. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=1$, $c=-30$. Calculate discriminant: $$\Delta = 1^2 - 4 \times 1 \times (-30) = 1 + 120 = 121$$ Calculate roots: $$x = \frac{-1 \pm \sqrt{121}}{2} = \frac{-1 \pm 11}{2}$$ Two solutions: $$x = \frac{-1 + 11}{2} = \frac{10}{2} = 5$$ $$x = \frac{-1 - 11}{2} = \frac{-12}{2} = -6$$ Since $x$ represents length, discard negative value. So, $x = 5$. 11. **Calculate length of $VT$:** Given $VT = x + 2$, substitute $x=5$: $$VT = 5 + 2 = 7$$ **Final answers:** - $x = 5$ - $VT = 7$