Question
The figure below is a rectangle. If BD = x+20 and AC = 4x+14, what is the total length of AC?
(1 point)
AC =
(1 point)
AC =
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Answered by
GPT-4o mini
In a rectangle, the diagonals are equal in length. Therefore, we can set the lengths of the diagonals equal to each other:
\[ BD = AC \]
Given the expressions for the lengths:
- \( BD = x + 20 \)
- \( AC = 4x + 14 \)
We can set up the equation:
\[ x + 20 = 4x + 14 \]
Next, let's solve for \( x \):
1. Subtract \( x \) from both sides:
\[ 20 = 3x + 14 \]
2. Subtract 14 from both sides:
\[ 6 = 3x \]
3. Divide by 3:
\[ x = 2 \]
Now that we have \( x \), we can find the length of \( AC \):
\[ AC = 4x + 14 \]
Substituting \( x = 2 \):
\[ AC = 4(2) + 14 = 8 + 14 = 22 \]
Thus, the total length of \( AC \) is:
**AC = 22**
\[ BD = AC \]
Given the expressions for the lengths:
- \( BD = x + 20 \)
- \( AC = 4x + 14 \)
We can set up the equation:
\[ x + 20 = 4x + 14 \]
Next, let's solve for \( x \):
1. Subtract \( x \) from both sides:
\[ 20 = 3x + 14 \]
2. Subtract 14 from both sides:
\[ 6 = 3x \]
3. Divide by 3:
\[ x = 2 \]
Now that we have \( x \), we can find the length of \( AC \):
\[ AC = 4x + 14 \]
Substituting \( x = 2 \):
\[ AC = 4(2) + 14 = 8 + 14 = 22 \]
Thus, the total length of \( AC \) is:
**AC = 22**
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