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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 BCDA
, AE¯¯¯¯¯¯¯¯
can be expressed as 3x
and ED¯¯¯¯¯¯¯¯
as −4x+21
. Identify a pattern to find the total length of BD¯¯¯¯¯¯¯¯
.

(1 point)
Responses

18
18

27
27

3
3

9

Answer 1

To find the total length of \( BD \), we can use the information given about the segments \( AE \) and \( ED \).

We know:

• \( AE = 3x \)
• \( ED = -4x + 21 \)

Since \( E \) lies on segment \( BD \), we can express the total length \( BD \) as the sum of \( AE \) and \( ED \):

\[ BD = AE + ED = 3x + (-4x + 21) \]

Simplifying this expression, we have:

\[ BD = 3x - 4x + 21 = -x + 21 \]

Next, we need to determine the value of \( x \). However, we are not provided any additional information that specifically gives us the value of \( x \). Therefore, we can examine the possible options for the total length of \( BD \):

1. Let’s evaluate the given responses:

   • If we let \( -x + 21 = 18 \):

     • \( -x = 18 - 21 \)
     • \( -x = -3 \)
     • \( x = 3 \)

   • If we let \( -x + 21 = 27 \):

     • \( -x = 27 - 21 \)
     • \( -x = 6 \)
     • \( x = -6 \) (not valid since \( x \) must be positive in a length context)

   • If we let \( -x + 21 = 3 \):

     • \( -x = 3 - 21 \)
     • \( -x = -18 \)
     • \( x = 18 \)

   • If we let \( -x + 21 = 9 \):

     • \( -x = 9 - 21 \)
     • \( -x = -12 \)
     • \( x = 12 \)

The only sensible findings for \( x \) lead us to \( x = 3 \), which corresponds to the \( BD \) total being:

\[ BD = -3 + 21 = 18 \]

Thus, the total length of \( BD \) is:

\[ \boxed{18} \]