Question

select the correct answer. abcd is a rectangle. ae = 3x - 6 and bd = 2x. what is ac? a. 24 b. 12 c. 6 d. 15

Explanation:

Step1: Recall rectangle property

In a rectangle, the diagonals are equal and bisect each other. So $AC = BD$ and $AE=\frac{1}{2}AC$. Since $AC = BD$, and $AE=\frac{1}{2}AC$, we have $AE=\frac{1}{2}BD$.

Step2: Set up equation

Given $AE = 3x - 6$ and $BD = 2x$, we substitute into $AE=\frac{1}{2}BD$. So $3x - 6=\frac{1}{2}(2x)$.

Step3: Solve the equation

First, simplify the right - hand side of the equation: $3x - 6=x$. Then subtract $x$ from both sides: $3x - x-6=x - x$, which gives $2x-6 = 0$. Add 6 to both sides: $2x=6$. Divide both sides by 2: $x = 3$.

Step4: Find length of $AC$

Since $AC = BD$ and $BD = 2x$, substitute $x = 3$ into the equation for $BD$. So $AC=2x=2\times3 = 6$.

Answer:

C. 6