Question

using the parallelogram diagonal theorem
what is the length of segment eb in parallelogram abcd?
30 units
34.5 units
69 units
138 units

Explanation:

Step1: Apply Parallelogram Diagonal Theorem

In a parallelogram, the diagonals bisect each other. So, \( AE = EB \) and \( DE = EC \). Also, \( DE = 2x + 9 \) and \( EB = 3x - 21 \), but since diagonals bisect each other, \( DE = EB \)? Wait, no, actually, in parallelogram \( ABCD \), diagonals \( AC \) and \( BD \) intersect at \( E \), so \( AE = EC \) and \( BE = ED \). Wait, the segments given are \( DE = 2x + 9 \) and \( EB = 3x - 21 \). Since \( BE = ED \) (because diagonals bisect each other), we set \( 2x + 9 = 3x - 21 \).

Step2: Solve for \( x \)

\( 2x + 9 = 3x - 21 \)
Subtract \( 2x \) from both sides: \( 9 = x - 21 \)
Add 21 to both sides: \( x = 9 + 21 = 30 \)

Step3: Find length of \( EB \)

Substitute \( x = 30 \) into \( EB = 3x - 21 \):
\( EB = 3(30) - 21 = 90 - 21 = 69 \)? Wait, no, wait, maybe I mixed up the segments. Wait, the diagram: \( DE = 2x + 9 \), \( EB = 3x - 21 \). But in parallelogram, diagonals bisect each other, so \( DE = EB \)? Wait, no, \( BD \) is a diagonal, so \( E \) is the midpoint, so \( DE = EB \). Wait, but when we solved \( 2x + 9 = 3x - 21 \), we got \( x = 30 \), then \( EB = 3(30) - 21 = 69 \)? But the options include 69. Wait, but let me check again. Wait, maybe the segments are \( AE = 2x + 9 \) and \( EB = 3x - 21 \), but in parallelogram, diagonals bisect each other, so \( AE = EC \) and \( BE = ED \), but if \( AC \) and \( BD \) intersect at \( E \), then \( AE = EC \) and \( BE = ED \), but the segments given are \( DE = 2x + 9 \) and \( EB = 3x - 21 \), so \( DE = EB \), so \( 2x + 9 = 3x - 21 \), \( x = 30 \), then \( EB = 3(30) - 21 = 69 \). So the length of \( EB \) is 69 units? Wait, but let me check the options: 30, 34.5, 69, 138. So 69 is an option. Wait, but maybe I made a mistake. Wait, no, let's re-express. Wait, maybe the segments are \( AE = 2x + 9 \) and \( EB = 3x - 21 \), but in parallelogram, diagonals bisect each other, so \( AE = EB \)? No, that's not correct. Wait, no, in parallelogram, diagonals bisect each other, so \( AE = EC \) and \( BE = ED \). So if \( DE = 2x + 9 \) and \( EB = 3x - 21 \), then \( DE = EB \), so \( 2x + 9 = 3x - 21 \), \( x = 30 \), \( EB = 3*30 -21 = 69 \). So the answer is 69 units.

Answer:

69 units