Use the properties of similar triangles to find expressions for $c$ and $d$. Start with the ratio: $\(\frac{(c+d)}{c}=\frac{h}{\frac{h}{2}}\)$ Similar triangle theorem


$$
\(\begin{array}{ll}
\frac{(c+d)}{c}=2 & \text { Simplify the right side } \\
c+d=2 c & \text { Multiply both sides by } c \\
d=c & \text { Subtract } c \text { from both sides }
\end{array}\)
$$


Now relate $c$ and $d$ to $b$ :


\(\begin{tabular}{ll}
$c+d=b$ & Shown on the diagram \\
$c+c=b$ & Substitute from above \\
$2 c=b$ & Simplify \\
$c=\frac{b}{2}$ & Express $c$ in terms of $b$
\end{tabular}\)

You have now demonstrated that the other side of the rectangle is equal to $\(\frac{b}{2}\)$. Next, put together the area formulas:
Area of the triangle $\(=\frac{1}{2} b h\)$
Area of the rectangle $\(=\frac{h}{2} \times \frac{b}{2}=\frac{1}{4} b h\)$

$$
\(\begin{array}{rlrl}
\text { Shaded Area } & =\text { Area of Triangle }- \text { Area of Rectangle } & & \text { Formula } \\
& =\frac{1}{2} b h-\frac{1}{4} b h & & \text { Substitute values } \\
& =\frac{1}{4} b h &
\end{array}\)
$$


Therefore the area of the rectangle equals the shaded area. Quantity A is equal to Quantity B, so the correct answer is $\(\mathbf{C}\)$.