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)
012
01
08
04

Answer 1

To solve for the length of diagonal BD in the kite ABCD, we start by analyzing the lengths given in the problem.

We know:
- \( BE = x + 3 \)
- \( DE = 4x \)

Since E is the intersection point of the diagonals in a kite, we can use the property of kites that states the diagonals are perpendicular and one diagonal bisects the other. From this, we can conclude:

\[
BE + DE = BD
\]

Substituting the expressions we have for BE and DE:

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

Now we can try to find specific values for \( x \) and the length of \( BD \). Since we need to find the total length of BD, we look for integer values for \( x \) that would make sense in the context of kite geometry.

We do this by considering the lengths:
\[
BE = x + 3 > 0 \implies x > -3
\]
\[
DE = 4x > 0 \implies x > 0
\]

So we conclude \( x > 0 \). Therefore, we can substitute different positive integer values for \( x \) to see if any of the total lengths \( BD\) ( \( 5x + 3 \) ) correspond to the provided options: 12, 01, 08, 04.

Let’s try some positive integers for \( x \):

1. For \( x = 1 \):
\[
BD = 5(1) + 3 = 5 + 3 = 8
\]

2. For \( x = 2 \):
\[
BD = 5(2) + 3 = 10 + 3 = 13
\]

3. For \( x = 3 \):
\[
BD = 5(3) + 3 = 15 + 3 = 18
\]

The only value that corresponds to one of the options provided is when \( x = 1 \), which gives:
\[
BD = 8
\]

From this investigation, the conjecture we can make is that for the dimensions provided in kite ABCD, the total length of diagonal \( BD \) is:

\[
\boxed{8}
\]