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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 answer

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:

\[ 20 = 3x + 14 \]

  1. Subtract 14 from both sides:

\[ 6 = 3x \]

  1. 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