Parallelogram Proofs Practice

Question

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Question
Use the image to answer the question.

A quadrilateral is labeled upper A upper B upper C upper D. A segment connects upper B and upper D. Another segment connects upper A and upper C. The point where the segments intersect is upper E.

In the rectangle, AC¯¯¯¯¯¯¯¯
has a length of 4x
and BD¯¯¯¯¯¯¯¯
has a length of 3x+6
. Using what you know about quadrilaterals and their diagonals, find the length of ED¯¯¯¯¯¯¯¯
.

(1 point)
ED¯¯¯¯¯¯¯¯=

Answer 1

In a parallelogram, the diagonals bisect each other. This means that the point where the diagonals intersect (point E in your diagram) divides each diagonal into two equal segments.

Given:

• Length of diagonal \( AC \) is \( 4x \).
• Length of diagonal \( BD \) is \( 3x + 6 \).

Since the diagonals bisect each other:

• \( AE = \frac{AC}{2} = \frac{4x}{2} = 2x \)
• \( BE = \frac{BD}{2} = \frac{3x + 6}{2} \)

In a parallelogram, \( AE \) and \( BE \) denote the half lengths of the diagonals. Since \( E \) is the midpoint for both diagonals, we have:

1. The length of \( ED \) (which is half of \( BD \)) can be found as follows:

\[ ED = \frac{BD}{2} = \frac{3x + 6}{2} \]

Thus, the answer is:

\[ ED = \frac{3x + 6}{2} \]

If you need to further simplify or evaluate \( ED \), you'll need a specific value for \( x \). If \( x \) is specified, just substitute it into the equation. If \( x \) is not specified, you can leave it in this form.