Answer:
A. BE = DE = 16
B. rectangle
Explanation:
The diagonals of a parallelogram bisect each other. The diagonals of a rectangle are the same length.
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A. diagonal BD
Point E is the midpoint of diagonal BD of the parallelogram, so we have ...
BE = DE
x² -48 = 2x
x² -2x -48 = 0 . . . . . put in standard form
(x -8)(x +6) = 0 . . . . factor
x = 8 or -6 . . . . . . values that satisfy the equation
We know the lengths have positive values, so the only value of x that works in this problem is x=8. Then the halves of diagonal BD are ...
BE = DE = 2x = 2(8)
BE = DE = 16
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B. comparison of diagonals
We know also that E is the midpoint of diagonal AC, so its length is ...
AC = 2×AE = 2×16 = 32
From the above, we know that ...
BD = BE +DE = 16 +16 = 32
The two diagonals are the same length, so the figure is a rectangle.
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Additional comment
If we knew something about the side lengths or the angle at which the diagonals cross, we could further classify the figure. Congruent adjacent sides, or perpendicular diagonals would mean this rectangle is a square.