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)
A. 4
B. 1
C. 8
D. 12

Answer 1

To find the total length of diagonal BD in kite ABCD, we start with the segments BE and DE given as follows:

- \( BE = x + 3 \)
- \( DE = 4x \)

Since E is the intersection point of diagonals AC and BD in a kite, we can set up the total length of diagonal BD as:

\[
BD = BE + DE
\]

Substituting the expressions for BE and DE, we have:

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

Now we need to investigate the values of \(x\) that make this form a valid length. Since BD is the shortest diagonal in the kite, it must be positive. Thus:

\[
5x + 3 > 0
\]

Solving for \(x\):

\[
5x > -3 \quad \Rightarrow \quad x > -\frac{3}{5}
\]

Next, we look at the individual segments \(BE = x + 3\) and \(DE = 4x\):

1. For \(BE\) to be positive:
\[
x + 3 > 0 \quad \Rightarrow \quad x > -3
\]

2. For \(DE\) to be positive:
\[
4x > 0 \quad \Rightarrow \quad x > 0
\]

The strongest condition is \(x > 0\).

Next, let's test some integer values for \(x\) to find the possible lengths of BD.

- If \(x = 1\):
- \(BE = 1 + 3 = 4\)
- \(DE = 4 \times 1 = 4\)
- So, \(BD = BE + DE = 4 + 4 = 8\).

- If \(x = 2\):
- \(BE = 2 + 3 = 5\)
- \(DE = 4 \times 2 = 8\)
- So, \(BD = BE + DE = 5 + 8 = 13\).

- If \(x = 0.5\):
- \(BE = 0.5 + 3 = 3.5\)
- \(DE = 4 \times 0.5 = 2\)
- So, \(BD = BE + DE = 3.5 + 2 = 5.5\).

Since \(BD\) is allowed to take various values, the operations we performed lead us to see different potential outcomes based on \(x\).

However, looking back at the options given:
A. 4
B. 1
C. 8
D. 12

The only confirmed integer value among our calculations for \(BD\) is 8 when \(x = 1\).

Hence, the conjectured total length of diagonal \(BD\) in kite ABCD is:

\[
\boxed{8}
\]