Question
The figure shown is a rectangle. If BD = x + 15 and AC = 3x + 1, what is the total length of AC?
Answers
Answer
Given the rectangle ABCD as shown, find the value of x if BE = 4x + 3 and EC = 10x - 15.
Answered by
GPT-4o mini
In a rectangle, the diagonals are equal in length. Therefore, we can set the lengths of the two diagonals \( BD \) and \( AC \) equal to each other:
\[
BD = AC
\]
Given the expressions for the lengths:
\[
x + 15 = 3x + 1
\]
Now, let's solve for \( x \):
1. Subtract \( x \) from both sides:
\[
15 = 2x + 1
\]
2. Subtract 1 from both sides:
\[
14 = 2x
\]
3. Divide both sides by 2:
\[
x = 7
\]
Now that we have the value of \( x \), we can substitute it back into the expression for \( AC \):
\[
AC = 3x + 1
\]
Substituting \( x = 7 \):
\[
AC = 3(7) + 1 = 21 + 1 = 22
\]
Thus, the total length of \( AC \) is:
\[
\boxed{22}
\]
\[
BD = AC
\]
Given the expressions for the lengths:
\[
x + 15 = 3x + 1
\]
Now, let's solve for \( x \):
1. Subtract \( x \) from both sides:
\[
15 = 2x + 1
\]
2. Subtract 1 from both sides:
\[
14 = 2x
\]
3. Divide both sides by 2:
\[
x = 7
\]
Now that we have the value of \( x \), we can substitute it back into the expression for \( AC \):
\[
AC = 3x + 1
\]
Substituting \( x = 7 \):
\[
AC = 3(7) + 1 = 21 + 1 = 22
\]
Thus, the total length of \( AC \) is:
\[
\boxed{22}
\]
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