In the rectangular coordinate plane, a parallelogram whose vertices are the intersection points of the lines $y = 2$ and $y = 6$ with the line $y = 2x$ has diagonals intersecting at the point $(0,4)$.
What is the area of this parallelogram in square units?
A) 16 B) 18 C) 20 D) 22 E) 24

In the rectangular coordinate plane, a square $ABCD$ with two vertices at $A(0, a)$ and $B(0, b)$ is given. The vertex $C$ of square $ABCD$ lies on the line $y = \frac{x}{3}$. If $a + b = 15$, what is the sum of the coordinates of point $D$?
A) 14
B) 18
C) 21
D) 24
E) 27

5 identical isosceles right triangles with right-angled side lengths of 1 unit are arranged as shown in Figure 1 such that their hypotenuses are on the same line and the vertices of adjacent triangles coincide.
Then triangle $ABC$ is rotated around point $A$ by some amount, and as shown in Figure 2, points B, C, and D become collinear.
Accordingly, what is the distance between points C and D in the final position in units?
A) 4
B) 5
C) 6
D) $3\sqrt{2}$
E) $4\sqrt{2}$

In the rectangular coordinate plane below, a red square with one side on the $y$-axis and a blue square with one side on the $x$-axis share a common vertex.
One vertex of each of the red and blue squares lies on the line $\dfrac{x}{2} + \dfrac{y}{3} = 1$.
According to this, what is the side length of the red square in units?
A) $\dfrac{14}{15}$ B) $\dfrac{15}{16}$ C) $\dfrac{16}{17}$ D) $\dfrac{17}{18}$ E) $\dfrac{18}{19}$

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the region between the lines $y = -\sqrt{3}x$ and $y = ax + b$ and the $x$-axis forms an equilateral triangle with area $9\sqrt{3}$ square units.
Accordingly, what is the product $a \cdot b$?
A) 18 B) 24 C) 27 D) 30 E) 36