Kite ABCD

Question

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

8
8

1
1

4
4

12

Answer 1

In a kite, the diagonals have specific properties. In particular, one diagonal (the shorter one) bisects the other diagonal at right angles.

Given:

• \( BE = x + 3 \)
• \( DE = 4x \)

Since \( E \) is the midpoint of diagonal \( BD \) in the kite, we can set up the equation:

\[ BE + DE = BD \]

Since \( BE = x + 3 \) and \( DE = 4x \):

\[ BD = (x + 3) + (4x) = 5x + 3 \]

To find the total length of diagonal \( BD \), we also need to recognize that the segments \( BE \) and \( DE \) are equal to each other because \( E \) is the midpoint:

\[ BE = DE \]

So we set the two expressions equal to each other:

\[ x + 3 = 4x \]

Now we can solve for \( x \):

\[ 3 = 4x - x \] \[ 3 = 3x \] \[ x = 1 \]

Now that we have \( x \), we can substitute it back into the equations for \( BE \) and \( DE \):

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

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

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

So, the total length of diagonal \( BD \) is \( \boxed{8} \).