________ The area of $\triangle{ABC}$ is $1$. $BD:DC=2:1$, and $E$ is the midpoint of $AC$. $AD$ intersects $BE$ at point $P$. Find the area of quadrilateral $PDCE$.
________ Rectangle $ABCD$ is divided into 4 smaller regions, $K$, $L$, $M$, and $N$, with equal area. If the ratio of the length and the height of rectangular region $K$ is $\dfrac{3}{2}$, what is the ratio of the length and the height of rectangular region $N$? Express your answer as a common fraction.
________ A rectangle metal sheet is cut into two circles and one smaller rectangle, as shown in shared regions, to form a cylinder-shaped oil container. Find the volume of the oil container in liters. Assume $\pi=3.14$, and round your answer to the nearest integer.
________ $\triangle{ABC}$ is right and $AC=3$, $BC=4$, $AB=5$. Then $AC$ is folded to $AB$. Find the area of the shaded region, which is not overlapping with other parts of the original triangle.
________ The area of the right $\triangle{ABC}$ is $12$. A semi-circle is drawn as shown. The area of the shaded region can be expressed as $a\pi-b$. Find the value of $a+b$.
________ The area of rectangle $ABCD$ is $36\ cm^2$. $E$, $F$, and $G$ are midpoint of $AB$, $BC$, and $CD$ respectively. $H$ is on $AD$. Find the area of the shaded region in $cm^2$.
________ The area of $\triangle{ABC}$ is $10\ cm^2$. $BA$ is extended to $D$ so that $DB=AB$. $CB$ is extended to $E$ so that $EA=2 AC$. $CB$ extended to $F$ so that $FB=3 BC$. Find the area of $\triangle{DEF}$ in $cm^2$.
________ The area of square $ABCD$ is $900\ cm^2$. $E$ is the midpoint of $CD$. $F$ is the midpoint of $BC$. $BE$ intersects with $DF$ at $M$, with $AF$ at $N$. Find the area of $\triangle{MNF}$ in $cm^2$.
________ $ABCD$ and $CGEF$ are two squares. $CF=3\ CH$. The area of $CHG$ is $6\ cm^2$. $E$ Find the area of pentagon $ABGEF$ in $cm^2$.
________ In trapezoid $ABCD$, $AD\parallel BC$. $AD=BE=EC$. $BD$ intersects with $AC$ at $O$, and $AE$ at $P$. The area of $\triangle{AOD}$ is $10\ cm^2$. Find the area of quadrilateral $OPEC$ in $cm^2$.
