Final answer:
By using the centroid property that divides medians in a 2:1 ratio, we can determine that GH equals 26 by solving the equation 2(GH) = EG.
Step-by-step explanation:
In ΔDEF with centroid G and median EH, the centroid divides the median into two segments with a ratio of 2:1, with the longer segment being adjacent to the vertex of the triangle. Given that EG = 3x+4 and GH = x+10, we can use this ratio to solve for x.
We know that EG is twice the length of GH in a triangle with a centroid, so we can set up the equation 2(GH) = EG, which yields 2(x+10) = 3x+4. Upon solving this equation, we can determine the value of x and consequently find the lengths of segments EG and GH.
Let's solve for x:
- 2(x+10) = 3x+4
- 2x + 20 = 3x + 4
- 20 - 4 = x
- x = 16
Therefore, GH = 16 + 10 = 26.