1. **Problem statement:** Find the value of $x$ where $IJ = x$, $HG = 8$, and $EF = 12$ in trapezoid $FEGH$ with median $JI$. 2. **Formula:** The median of a trapezoid is the segment connecting the midpoints of the legs and is parallel to the bases. Its length is the average of the lengths of the two bases: $$JI = \frac{HG + EF}{2}$$ 3. **Apply the formula:** $$x = \frac{8 + 12}{2}$$ 4. **Calculate:** $$x = \frac{20}{2} = 10$$ 5. **Answer:** The value of $x$ is $10$. --- 1. **Problem statement:** Find $x$ and $JI$ when $HG = 14$, $EF = 18$, and $IJ = x$. 2. **Formula:** Same as above: $$JI = \frac{HG + EF}{2}$$ 3. **Apply the formula:** $$x = \frac{14 + 18}{2}$$ 4. **Calculate:** $$x = \frac{32}{2} = 16$$ 5. **Answer:** $x = 16$ and $JI = 16$. --- 1. **Problem statement:** Find $x$ when $HG = x$, $JI = 16$, and $EF = 22$. 2. **Formula:** $$JI = \frac{HG + EF}{2}$$ 3. **Substitute known values:** $$16 = \frac{x + 22}{2}$$ 4. **Multiply both sides by 2:** $$2 \times 16 = x + 22$$ $$\cancel{2} \times 16 = x + 22$$ $$32 = x + 22$$ 5. **Solve for $x$:** $$x = 32 - 22 = 10$$ 6. **Answer:** $x = 10$. --- 1. **Problem statement:** Find $y$ and $HG$ when $HG = y - 2$, $JI = 20$, and $EF = 31$. 2. **Formula:** $$JI = \frac{HG + EF}{2}$$ 3. **Substitute known values:** $$20 = \frac{(y - 2) + 31}{2}$$ 4. **Multiply both sides by 2:** $$2 \times 20 = y - 2 + 31$$ $$40 = y + 29$$ 5. **Solve for $y$:** $$y = 40 - 29 = 11$$ 6. **Find $HG$:** $$HG = y - 2 = 11 - 2 = 9$$ 7. **Answer:** $y = 11$ and $HG = 9$. --- 1. **Problem statement:** Find $x$ and $IE$ when $HE = 10$ and $IE = x - 4$. 2. **Note:** Since $HE$ and $IE$ are parts of the trapezoid, and no further info is given, assume $HE = IE$ for equality. 3. **Set equal:** $$10 = x - 4$$ 4. **Solve for $x$:** $$x = 10 + 4 = 14$$ 5. **Find $IE$:** $$IE = 14 - 4 = 10$$ 6. **Answer:** $x = 14$ and $IE = 10$. --- **Summary:** - 1) $x = 10$ - 2) $x = 16$, $JI = 16$ - 3) $x = 10$ - 4) $y = 11$, $HG = 9$ - 5) $x = 14$, $IE = 10$