Parallelogram Proofs Quick Check

Question

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Question
Kite ABCD
has diagonal lines AC¯¯¯¯¯¯¯¯
and BD¯¯¯¯¯¯¯¯
that intersect at E
, with BD¯¯¯¯¯¯¯¯
being the shortest diagonal line. Given that BE¯¯¯¯¯¯¯¯=x+3
and DE¯¯¯¯¯¯¯¯=4x
, investigate the pattern and make a conjecture about line BD¯¯¯¯¯¯¯¯
. What is the total length of BD¯¯¯¯¯¯¯¯
?(1 point)
Responses

4
4

1
1

12
12

8

Answer 1

To find the total length of diagonal \( BD \) in kite \( ABCD \), we know that the diagonals of a kite intersect at right angles and the shorter diagonal is bisected by the longer diagonal. Since \( BE \) and \( DE \) are segments of the diagonal \( BD \) with \( BE = x + 3 \) and \( DE = 4x \), we can express the total length of \( BD \) as:

\[ BD = BE + DE = (x + 3) + (4x) \]

This simplifies to:

\[ BD = 5x + 3 \]

To find the total length of \( BD \), we need to set the two segments equal, since point \( E \) is the midpoint of \( BD \). This means:

\[ BE = DE \] which gives us:

\[ x + 3 = 4x \]

Solving for \( x \):

1. Subtract \( x \) from both sides: \[ 3 = 3x \]
2. Divide by 3: \[ x = 1 \]

Now, substituting back into \( BE \) and \( DE \):

• \( BE = x + 3 = 1 + 3 = 4 \)
• \( DE = 4x = 4 \cdot 1 = 4 \)

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

\[ BD = BE + DE = 4 + 4 = 8 \]

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

8