1. **Problem Statement:** Find the value of $x$ in a triangle where a smaller triangle is formed by connecting the midpoints of the sides of a larger triangle. The larger triangle has sides 76 and 98, and the smaller triangle has one side labeled 44 and another side labeled $x$. 2. **Key Concept:** The triangle formed by joining the midpoints of the sides of a triangle is called the medial triangle. Each side of the medial triangle is parallel to one side of the original triangle and exactly half its length. 3. **Formula:** If the side of the larger triangle is $S$, then the corresponding side of the medial triangle is $\frac{S}{2}$. 4. **Given:** - Larger triangle sides: 76 and 98 - Smaller triangle side: 44 and $x$ 5. **Step 1: Find the side corresponding to 44 in the larger triangle.** Since 44 is a side of the smaller triangle, it must be half of the corresponding side in the larger triangle. $$ 44 = \frac{S}{2} \implies S = 44 \times 2 = 88 $$ 6. **Step 2: Find $x$ using the same rule.** The side labeled $x$ in the smaller triangle corresponds to a side in the larger triangle, which is either 76 or 98. Since 44 corresponds to 88, and 88 is not 76 or 98, the side 44 corresponds to a side of length 88 in the larger triangle (not given explicitly). Therefore, $x$ corresponds to one of the given sides 76 or 98. 7. **Step 3: Calculate $x$ as half of the corresponding larger triangle side.** We check both possibilities: - If $x$ corresponds to 76, then $x = \frac{76}{2} = 38$ - If $x$ corresponds to 98, then $x = \frac{98}{2} = 49$ 8. **Step 4: Determine which side $x$ corresponds to.** Since the smaller triangle has sides 44 and $x$, and 44 corresponds to 88, the other side $x$ must correspond to either 76 or 98. Given the problem setup, the side $x$ corresponds to 98, so: $$ x = \frac{98}{2} = 49 $$ **Final answer:** $$ \boxed{49} $$