Question

the area of rhombus abcd is 120 square units. ae = 12 and bd = x - 2. what is the value of x and the length of segment bd? x = bd = units

Explanation:

Step1: Recall the area formula of a rhombus

The area of a rhombus $A=\frac{1}{2}\times d_1\times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals. In rhombus $ABCD$, let $AC$ and $BD$ be the diagonals, and $AC = 2AE$. Since $AE = 12$, then $AC=24$. Let $BD=x - 2$. The area $A = 120$.

Step2: Substitute values into the area formula

We know that $A=\frac{1}{2}\times AC\times BD$. Substituting $A = 120$, $AC = 24$ and $BD=x - 2$ into the formula, we get $120=\frac{1}{2}\times24\times(x - 2)$.

Step3: Solve the equation for $x$

First, simplify the right - hand side of the equation: $\frac{1}{2}\times24\times(x - 2)=12\times(x - 2)=12x-24$. So the equation becomes $120=12x - 24$. Add 24 to both sides: $120 + 24=12x$, i.e., $144 = 12x$. Divide both sides by 12, we get $x = 12$.

Step4: Find the length of $BD$

Since $BD=x - 2$ and $x = 12$, then $BD=12-2=10$.

Answer:

$x = 12$
$BD = 10$ units