Question

In the rectangle BCDA, AE can be expressed as 3x and ED as -4x+21. Identify a pattern to find the total length of BD.
(1 point)
A. 27
B. 18
C. 3
D. 9

Answer 1

To find the total length of BD in rectangle BCDA with sides AE and ED, we can sum the lengths of AE and ED.

Let's express the total length BD:

1. **AE = 3x**
2. **ED = -4x + 21**

We need to find the total length of BD, which is the sum of AE and ED:

\[
BD = AE + ED = 3x + (-4x + 21)
\]

Now simplify the expression:

\[
BD = 3x - 4x + 21
\]
\[
BD = -x + 21
\]

To determine the potential lengths given as options (A. 27, B. 18, C. 3, D. 9), we can set \( -x + 21 \) equal to each of the options and solve for \( x \):

1. **Option A: \( -x + 21 = 27 \)**
\[
-x = 27 - 21
\]
\[
-x = 6 \quad \Rightarrow \quad x = -6
\]

2. **Option B: \( -x + 21 = 18 \)**
\[
-x = 18 - 21
\]
\[
-x = -3 \quad \Rightarrow \quad x = 3
\]

3. **Option C: \( -x + 21 = 3 \)**
\[
-x = 3 - 21
\]
\[
-x = -18 \quad \Rightarrow \quad x = 18
\]

4. **Option D: \( -x + 21 = 9 \)**
\[
-x = 9 - 21
\]
\[
-x = -12 \quad \Rightarrow \quad x = 12
\]

Now we can check the possible \( x \) values just calculated, but since none of the resulting \( x \) values correlate directly with the options we started with, we should focus on determining if specific values of \( x \) yield integer lengths for BD.

1. If \( x = 3 \):
\[
BD = -3 + 21 = 18
\] (Option B)

Thus, the value of \( BD \) that fits the options is **B. 18**.

Therefore, the correct answer is:

**B. 18**.