Question

For context, it shows a rectangle, with the bottom left point being A, top left being B, top right being C, bottom right being D, and 2 lines going diagonally across with point E in the middle

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

ED=____

Answer 1

In a rectangle, the diagonals are equal in length. This means that the lengths of diagonals AC and BD are the same.

Given:
- Length of diagonal AC = \(4x\)
- Length of diagonal BD = \(3x + 6\)

Setting these equal to each other gives us the equation:

\[
4x = 3x + 6
\]

To solve for \(x\), we can subtract \(3x\) from both sides:

\[
4x - 3x = 6
\]

This simplifies to:

\[
x = 6
\]

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

Substituting \(x = 6\) into the expression for diagonal AC:

\[
AC = 4x = 4(6) = 24
\]

Substituting \(x = 6\) into the expression for diagonal BD:

\[
BD = 3x + 6 = 3(6) + 6 = 18 + 6 = 24
\]

Both diagonals have a length of 24.

In a diagonal AC, the midpoint (point E) divides the diagonal into two equal parts. Since the length of AC is 24, the lengths of AE and EC are each:

\[
AE = EC = \frac{24}{2} = 12
\]

Since the rectangle is symmetrical, point D and point B will also be split in half along BD similarly. Therefore, the lengths of segments ED and EB are also equal:

\[
ED = EB
\]

Since \(BD\) is also \(24\):

\[
ED + EB = 24
\]

Thus we can conclude:

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

Therefore, the length of \(ED\) is:

\[
\boxed{12}
\]