Let m be a real number. In the rectangular coordinate plane,

• the slope of a line passing through the point $( 0,1 )$ is $m$,
• the slope of a line passing through the point $( 0,0 )$ is $2 m$,
• the slope of a line passing through the point $( 1,0 )$ is $3 m$, and these three lines intersect at one point.

Accordingly, what is the value of $m$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 5 }$
E) $\frac { 4 } { 5 }$

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

In the rectangular coordinate plane, it is known that a line $d$ passes through point $A(-4, 1)$ and is perpendicular to the line $2x - y = 5$. If the point where line $d$ intersects the x-axis is $(a, 0)$ and the point where it intersects the y-axis is $(0, b)$, what is the sum $a + b$?
A) -3
B) -1
C) 0
D) 1
E) 3

In the rectangular coordinate plane, points $A(2, 7)$ and $B(-1, 4)$ are translated 3 units in the positive direction along the x-axis to obtain points $D$ and $C$ respectively.
Accordingly, what is the area of the quadrilateral with vertices at points A, B, C, and D in square units?
A) 9
B) 10
C) 11
D) 12
E) 13

In the rectangular coordinate plane, one vertex of a triangle is at the origin, its centroid is at the point $( 0,6 )$, and its orthocenter is at the point $( 0,8 )$.
Accordingly, what is the area of this triangle in square units?
A) 18
B) 21
C) 24
D) 27
E) 30

In the rectangular coordinate plane, points A and B lie on the line $y = x + 2$, and the distance between them is 3 units.
Given that the coordinates of the midpoint of segment [AB] are $( -1, 1 )$, in which regions of the analytic plane are points A and B located?
A) Both in region II
B) Both in region III
C) One in region I, the other in region II
D) One in region I, the other in region III
E) One in region II, the other in region III

In the rectangular coordinate plane, two lines that intersect perpendicularly at point $A ( 3,4 )$ have slopes whose sum is $\frac { 3 } { 2 }$.
If the points where these two lines intersect the x-axis are points B and C, what is the area of triangle ABC in square units?
A) 24
B) 20
C) 16
D) 12
E) 8

In the rectangular coordinate plane, the symmetric point of $( 4,4 )$ with respect to a line passing through $( 1,0 )$ is $( a , 0 )$. Accordingly, what is the product of the values that $a$ can take?
A) $-24$
B) $-16$
C) $-8$
D) $16$
E) $32$

In the rectangular coordinate plane, the line $2x + y = 12$ and a line d intersect at point $\mathrm{A}(4,4)$. These two lines divide every circle centered at point $\mathrm{A}(4,4)$ into four equal areas.
Accordingly, which of the following is the equation of line d?
A) $-2x + y = -4$ B) $x - 3y = -8$ C) $3x + y = 16$ D) $x + 2y = 12$ E) $x - 2y = -4$

In a rectangular coordinate plane, points $A(9,2)$, $B(10,1)$, $C$, $D(4,13)$, $E(3,6)$ and $F$ are given.
Given that the centroid of triangle $ABC$ and the centroid of triangle $DEF$ are the same point, what is the distance between points $C$ and $F$ in units?
A) 10 B) 13 C) 15 D) 17 E) 20

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) $\dfrac{18}{19}$

In a rectangular coordinate plane, what is the area of the triangular region bounded by the lines $2x - y = 0$, $x + 2y = 0$ and $x - 8y + 30 = 0$ in square units?
A) 9 B) 12 C) 15 D) 18 E) 21

In a rectangular coordinate plane, point $A(a, b)$; its reflection with respect to point $B(3, 0)$ is point $C$, and its reflection with respect to the $y$-axis is point $D$.
Given that the equation of the line passing through points $C$ and $D$ is $y = -x - 1$, what is the sum $a + b$?
A) 7 B) 13 C) 15 D) 19 E) 24

Let $a$ and $b$ be positive real numbers. In the rectangular coordinate plane, the acute angles that the lines $d_{1}$ and $d_{2}$ shown make with the $x$-axis are $A$ and $B$ respectively, as shown in the figure.
Accordingly, which of the following is the expression for the ratio $\frac{a}{b}$ in terms of $A$ and $B$?
A) $\frac{\tan A}{\tan B}$ B) $\cot A \cdot \cot B$ C) $\cot A - \tan B$ D) $1 + \cot A \cdot \tan B$ E) $1 - \tan A \cdot \cot B$

In the rectangular coordinate plane, a triangle $OAB$ with one vertex at the origin and the other two vertices on the axes, and the line segment $[PR]$ connecting the points $P(6, -3)$ and $R(-2, 9)$ are drawn. The line segment $[PR]$ passes through the midpoints of both $[OA]$ and $[OB]$.
According to this, what is the area of triangle $OAB$ in square units?
A) 36 B) 42 C) 48 D) 54 E) 60

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

In the rectangular coordinate plane, when point $A$ is translated 15 units in the negative direction along the $x$-axis, the resulting point lies on the line $d: 4x - 3y + 24 = 0$.
Accordingly, if point $A$ is translated how many units in the positive direction along the $y$-axis, the resulting point will lie on line $d$?
A) 9 B) 12 C) 16 D) 20 E) 25