To investigate the pattern and solve for the total length of \(BD\), we need to find the total length using the given expressions for \(BE\) and \(DE\).
We have:
- \(BE = x + 3\)
- \(DE = 4x\)
Since point E is on line BD, we can express the total length of \(BD\) (denoted as \(BD\)) as the sum of \(BE\) and \(DE\):
\[ BD = BE + DE = (x + 3) + (4x) \]
Combining like terms, we get:
\[ BD = x + 3 + 4x = 5x + 3 \]
Next, we need additional information or constraints to determine the value of \(x\). In a kite, the diagonals are known to intersect at right angles, and since \(BD\) is the shorter diagonal, we can postulate a specific case. However, without loss of generality, if we have potential lengths from the given responses, we can test values to find \(x\).
The possible responses given were: 12, 12, 4, 4, 1, 1, 8.
We can test these values against the expression \(BD = 5x + 3\):
-
If \(BD = 12\): \[ 12 = 5x + 3 \] \[ 12 - 3 = 5x \] \[ 9 = 5x \] \[ x = \frac{9}{5} \] (not an integer)
-
If \(BD = 4\): \[ 4 = 5x + 3 \] \[ 4 - 3 = 5x \] \[ 1 = 5x \] \[ x = \frac{1}{5} \] (not an integer)
-
If \(BD = 1\): \[ 1 = 5x + 3 \] \[ 1 - 3 = 5x \] \[ -2 = 5x \] \[ x = -\frac{2}{5} \] (not valid)
-
If \(BD = 8\): \[ 8 = 5x + 3 \] \[ 8 - 3 = 5x \] \[ 5 = 5x \] \[ x = 1 \] (valid)
Now substituting \(x = 1\) into \(BE\) and \(DE\):
-
\(BE = 1 + 3 = 4\)
-
\(DE = 4(1) = 4\)
Therefore, \(BD = BE + DE = 4 + 4 = 8\).
Thus, we can conclude with our conjecture that the total length of diagonal \(BD\) is:
\[ \boxed{8} \]