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

1
1

4
4

8
8

12

Answer 1

In a kite, the diagonals intersect at right angles, and one of the diagonals bisects the other one. Here, we are given the lengths of segments \( BE \) and \( DE \) that are parts of the diagonal \( BD \).

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

Since diagonal \( BD \) is bisected at point \( E \), we have:
\[
BD = BE + DE = (x + 3) + (4x) = 5x + 3
\]

To make a conjecture about the total length of \( BD \), we can observe the conditions:

### Step 1: Set the equation for the total length.
We express \( BD \) in terms of \( x \):
\[
BD = 5x + 3
\]

### Step 2: Evaluate the possible responses.
Considering the response options given (1, 4, 8, 12), we can solve for \( x \) to evaluate if \( BD \) can take one of those values.

### Evaluating each option:
1. **If \( BD = 1 \):**
\[
5x + 3 = 1 \implies 5x = -2 \implies x = -\frac{2}{5} \quad (\text{not valid})
\]

2. **If \( BD = 4 \):**
\[
5x + 3 = 4 \implies 5x = 1 \implies x = \frac{1}{5}
\]

3. **If \( BD = 8 \):**
\[
5x + 3 = 8 \implies 5x = 5 \implies x = 1
\]

4. **If \( BD = 12 \):**
\[
5x + 3 = 12 \implies 5x = 9 \implies x = \frac{9}{5} \quad (\text{not valid to yield an integer value for DE})
\]

### Step 3: Checking the values of \( x \).
- For \( BD = 4 \):
- \( x = \frac{1}{5} \)
- \( BE = x + 3 = \frac{1}{5} + 3 = \frac{16}{5} \)
- \( DE = 4x = 4 \cdot \frac{1}{5} = \frac{4}{5} \)

Total \( BD = \frac{16}{5} + \frac{4}{5} = 4 \) (Valid)

- For \( BD = 8 \):
- \( x = 1 \)
- \( BE = x + 3 = 1 + 3 = 4 \)
- \( DE = 4x = 4 \cdot 1 = 4 \)

Total \( BD = 4 + 4 = 8 \) (Valid)

### Conclusion
The possible total lengths for \( BD \) from the responses given are valid for both 4 and 8. However, since 8 is also derived from \( x \) giving \( DE = 4 \), it makes it valid.

Thus, the total length of diagonal \( BD \) in kite \( ABCD \) can therefore be concluded as:

**The total length of \( BD \) is 8.**