In a rectangle, the diagonals are equal in length. This means that the lengths of diagonals AC and BD are the same.
Given:
- Length of diagonal AC = \(4x\)
- Length of diagonal BD = \(3x + 6\)
Setting these equal to each other gives us the equation:
\[ 4x = 3x + 6 \]
To solve for \(x\), we can subtract \(3x\) from both sides:
\[ 4x - 3x = 6 \]
This simplifies to:
\[ x = 6 \]
Now that we have \(x\), we can find the lengths of the diagonals:
Substituting \(x = 6\) into the expression for diagonal AC:
\[ AC = 4x = 4(6) = 24 \]
Substituting \(x = 6\) into the expression for diagonal BD:
\[ BD = 3x + 6 = 3(6) + 6 = 18 + 6 = 24 \]
Both diagonals have a length of 24.
In a diagonal AC, the midpoint (point E) divides the diagonal into two equal parts. Since the length of AC is 24, the lengths of AE and EC are each:
\[ AE = EC = \frac{24}{2} = 12 \]
Since the rectangle is symmetrical, point D and point B will also be split in half along BD similarly. Therefore, the lengths of segments ED and EB are also equal:
\[ ED = EB \]
Since \(BD\) is also \(24\):
\[ ED + EB = 24 \]
Thus we can conclude:
\[ ED = \frac{24}{2} = 12 \]
Therefore, the length of \(ED\) is:
\[ \boxed{12} \]
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