Question

select the correct answer.
abcd is a rectangle. ae = 3x - 6 and bd = 2x. what is ac?
image of rectangle abcd with diagonals ac and bd intersecting at e
a. 24
b. 15
c. 12
d. 6

Explanation:

Step1: Recall rectangle diagonal properties

In a rectangle, diagonals are equal and bisect each other. So \( AC = BD \) and \( AE=\frac{1}{2}AC \), \( BE = \frac{1}{2}BD \), and \( AC = 2AE \), \( BD=2AE \) (since \( AC = BD \)).
Given \( AE = 3x - 6 \) and \( BD=2x \), and since \( AC = BD \) and \( AC = 2AE \), we have \( BD=2AE \). Substitute the expressions: \( 2x=2(3x - 6) \).

Step2: Solve for \( x \)

Simplify the equation \( 2x = 2(3x - 6) \). Divide both sides by 2: \( x=3x - 6 \). Subtract \( x \) from both sides: \( 0 = 2x - 6 \). Add 6 to both sides: \( 2x=6 \), then \( x = 3 \)? Wait, no, wait. Wait, if \( AC = BD \) and \( AE=\frac{1}{2}AC \), so \( AC = 2AE \), and \( BD = AC \), so \( BD=2AE \). So \( 2x=2(3x - 6) \) → divide both sides by 2: \( x = 3x - 6 \) → \( - 2x=-6 \) → \( x = 3 \)? Wait, that can't be. Wait, maybe I made a mistake. Wait, in a rectangle, diagonals are equal and bisect each other, so \( AE = \frac{1}{2}AC \) and \( BD = AC \), so \( AE=\frac{1}{2}BD \). So \( 3x - 6=\frac{1}{2}(2x) \). Ah! That's the mistake. \( BD = 2x \), so \( \frac{1}{2}BD=x \), and \( AE=\frac{1}{2}AC=\frac{1}{2}BD \), so \( 3x - 6=x \). Now solve \( 3x - 6=x \). Subtract \( x \) from both sides: \( 2x - 6 = 0 \), add 6: \( 2x=6 \), \( x = 3 \)? No, wait, no. Wait, \( BD = 2x \), \( AC = BD \), so \( AC = 2x \), and \( AE=\frac{1}{2}AC \), so \( AE=x \). But \( AE = 3x - 6 \), so \( x=3x - 6 \), \( - 2x=-6 \), \( x = 3 \). Then \( BD = 2x=6 \), so \( AC = BD = 6 \)? But let's check again. Wait, if \( x = 3 \), \( AE=3(3)-6 = 3 \), then \( AC = 2AE = 6 \), and \( BD = 2x = 6 \), which matches. So then \( AC = 6 \)? But let's check the options. Option D is 6. Wait, but let's re - derive.

Correct property: In rectangle \( ABCD \), diagonals \( AC \) and \( BD \) bisect each other at \( E \), so \( AE = EC \) and \( BE = ED \), and \( AC = BD \). So \( AE=\frac{1}{2}AC \) and \( BD = AC \), so \( AE=\frac{1}{2}BD \). Given \( AE = 3x - 6 \) and \( BD = 2x \), so:

\( 3x - 6=\frac{1}{2}(2x) \)

Simplify right side: \( \frac{1}{2}(2x)=x \)

So equation: \( 3x - 6=x \)

Subtract \( x \) from both sides: \( 2x - 6 = 0 \)

Add 6 to both sides: \( 2x=6 \)

Divide by 2: \( x = 3 \)

Now, \( BD = 2x=6 \), and since \( AC = BD \) (diagonals of rectangle are equal), \( AC = 6 \).

Step3: Find \( AC \)

Since \( AC = BD \) and \( BD = 2x \), and we found \( x = 3 \), so \( BD = 2\times3 = 6 \), so \( AC = 6 \).

Answer:

D. 6