Answer
$BD = 1$
$AC = 1$
Work Step by Step
According to Theorem 6-15, the diagonals of a rectangle are congruent. Therefore, we can set $AC$ and $BD$, the diagonals of $ABCD$, equal to one another to solve for $c$:
$AC = BD$
Substitute with the expressions given for each diagonal:
$\frac{3c}{9} = 4 - c$
Multiply both sides of the equation by $9$ to get rid of the fraction:
$3c = 36 - 9c$
Add $9c$ to each side of the equation to move the variable to the left side of the equation:
$12c = 36$
Divide each side of the equation by $12$ to solve for $c$:
$c = 3$
Now that we have the value of $c$, we can plug $3$ in for $c$. We do not need to evaluate both expressions because we know that the lengths of the two diagonals are equal to one another:
$BD = 4 - 3$
Subtract to solve:
$BD = 1$
If $BD = 1$, then $AC = 1$ because diagonals of rectangles are congruent.
