In the rectangle, AC has a length of 4x and BD has a length of 3x + 6: Using what you know about quadrilaterals and their diagonals, find the length of ED.

Question

(1 point

Answer 1

In a rectangle, the diagonals are equal in length. Therefore, if we have diagonals AC and BD, we can set their lengths equal to each other.

Given:

• Length of diagonal AC = \( 4x \)
• Length of diagonal BD = \( 3x + 6 \)

We can set up the equation:

\[ 4x = 3x + 6 \]

Now, solve for \( x \):

1. Subtract \( 3x \) from both sides: \[ 4x - 3x = 6 \] \[ x = 6 \]

Now that we have the value of \( x \), we can find the lengths of the diagonals:

1. Substitute \( x = 6 \) into \( AC \): \[ AC = 4x = 4(6) = 24 \]

2. Substitute \( x = 6 \) into \( BD \): \[ BD = 3x + 6 = 3(6) + 6 = 18 + 6 = 24 \]

Both diagonals AC and BD equal 24, confirming that our calculations are correct.

Next, to find \( ED \), we must note that in a rectangle, the midpoint of one diagonal (let's say point E) and the midpoint of the other diagonal (point D) are equal to half the length of the diagonals from one another.

Since both diagonals have the same length, ED (which is half of the diagonal length of AC or BD) is:

\[ ED = \frac{AC}{2} = \frac{24}{2} = 12 \]

Thus, the length of ED is \( \boxed{12} \).