On the coordinate plane, there is a polygonal region $\Gamma$ (including boundary) as shown in the figure. If $k > 0$ , the line $7 x + 2 y = k$ and the two coordinate axes form a triangular region such that the polygonal region $\Gamma$ lies within this triangular region (including boundary), then the minimum positive real number $k =$ (7)(8).

On the coordinate plane, there is a trapezoid with four vertices at $A ( 0,0 ) , B ( 1,0 ) , P , Q$ , where the line passing through $P$ and $Q$ has equation $y = 2 x + 4$ . If the coordinates of point $Q$ are $( a , 2 a + 4 )$ , where $a \geq 0$ is a real number, then the area of trapezoid $A B P Q$ is (14)$a +$ (16). (Reduce to lowest terms)

On a coordinate plane, there are two points $A ( - 3,4 ) , B ( 3,2 )$ and a line $L$. Points $A$ and $B$ are on opposite sides of line $L$, and $\vec { n } = ( 4 , - 3 )$ is a normal vector to line $L$. The distance from point $A$ to line $L$ is 5 times the distance from point $B$ to line $L$. Based on the above, answer the following questions.
(1) Find the dot product of vector $\overrightarrow { A B }$ and vector $\vec { n }$. (4 points)
(2) Find the equation of line $L$. (4 points)
(3) Point $P$ is on line $L$ and $\overline { P A } = \overline { P B }$. Find the coordinates of point $P$. (4 points)

Two lines $L _ { 1 } , L _ { 2 }$ on the coordinate plane both have positive slopes, and the angle bisector of one of the angles formed by $L _ { 1 } , L _ { 2 }$ has slope $\frac { 11 } { 9 }$ . Another line $L$ passes through the point $( 2 , \frac { 1 } { 3 } )$ and forms a bounded region with $L _ { 1 } , L _ { 2 }$ that is an equilateral triangle. Which of the following options is the equation of $L$?
(1) $11 x - 9 y = 19$
(2) $9 x + 11 y = 25$
(3) $11 x + 9 y = 25$
(4) $27 x - 33 y = 43$
(5) $27 x + 33 y = 65$

There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options.
(1) The laser beam lies on a line passing through the origin with slope $a$
(2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$
(3) The player can hit all three disks with just one laser beam
(4) The player needs to fire at least three laser beams to hit all three disks
(5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$

Consider the line $L: 5y + (2k-4)x - 10k = 0$ on the coordinate plane (where $k$ is a real number), and the rectangle $OABC$ with vertices at $O(0,0)$, $A(10,0)$, $B(10,6)$, $C(0,6)$. Let $L$ intersect the line $OC$ and the line $AB$ at points $D$ and $E$ respectively. Select the correct options.
(1) When $k = 4$, the line $L$ passes through point $A$
(2) If the line $L$ passes through point $C$, then the slope of $L$ is $-\frac{5}{2}$
(3) If point $D$ is on the line segment $\overline{OC}$, then $0 \leq k \leq 3$
(4) If $k = \frac{1}{2}$, then the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary)
(5) If the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary), then the slope of $L$ could be $\frac{3}{10}$

In an open space, there are three utility poles perpendicular to the ground with equal heights and equally spaced bases on a straight line. A person uses one-point perspective to draw these three utility poles on a canvas. A coordinate system is set up on the canvas so that the utility poles are parallel to the $y$-axis. The three base points are $A_{1}(0,0)$, $A_{2}$, $A_{3}$, all on the line $L: x + 3y = 0$; the three top points are $B_{1}(0,3)$, $B_{2}$, $B_{3}$, all on the line $M: 2x - 3y + 9 = 0$, as shown in the figure. It is known that $\overline{A_{3}B_{3}} = 2\overline{A_{1}B_{1}}$, and by one-point perspective, the intersection of lines $A_{1}B_{3}$ and $A_{3}B_{1}$ lies on line $A_{2}B_{2}$. Let $P$ be the intersection of $L$ and $M$ (this point is also called the "vanishing point").
If $\overrightarrow{PA_{1}} = k\overrightarrow{PA_{3}}$, then the value of $k$ is $\square$. (Express as a fraction in lowest terms)

In an open space, there are three utility poles perpendicular to the ground with equal heights and equally spaced bases on a straight line. A person uses one-point perspective to draw these three utility poles on a canvas. A coordinate system is set up on the canvas so that the utility poles are parallel to the $y$-axis. The three base points are $A_{1}(0,0)$, $A_{2}$, $A_{3}$, all on the line $L: x + 3y = 0$; the three top points are $B_{1}(0,3)$, $B_{2}$, $B_{3}$, all on the line $M: 2x - 3y + 9 = 0$, as shown in the figure. It is known that $\overline{A_{3}B_{3}} = 2\overline{A_{1}B_{1}}$, and by one-point perspective, the intersection of lines $A_{1}B_{3}$ and $A_{3}B_{1}$ lies on line $A_{2}B_{2}$. Let $P$ be the intersection of $L$ and $M$ (this point is also called the "vanishing point").
Find the coordinates of points $P$ and $B_{3}$.

On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
(Continuing from question 19, where $\sin\theta = \frac{3}{5}$) Find the distance from point $A$ to line $BQ$, and find the area of quadrilateral $PABQ$. (Non-multiple choice question, 6 points)

In an open space, there are three utility poles perpendicular to the ground with equal heights and equally spaced bases on a straight line. A person uses one-point perspective to draw these three utility poles on a canvas. A coordinate system is set up on the canvas so that the utility poles are parallel to the $y$-axis. The three base points are $A_{1}(0,0)$, $A_{2}$, $A_{3}$, all on the line $L: x + 3y = 0$; the three top points are $B_{1}(0,3)$, $B_{2}$, $B_{3}$, all on the line $M: 2x - 3y + 9 = 0$, as shown in the figure. It is known that $\overline{A_{3}B_{3}} = 2\overline{A_{1}B_{1}}$, and by one-point perspective, the intersection of lines $A_{1}B_{3}$ and $A_{3}B_{1}$ lies on line $A_{2}B_{2}$. Let $P$ be the intersection of $L$ and $M$ (this point is also called the "vanishing point").
Suppose a bee stops on the middle utility pole at a position where the ratio of distances from the base to the top is $1:2$. The person wants to draw this bee on the line segment $A_{2}B_{2}$ on the canvas. Assuming the bee's position on the canvas is point $Q$, that is, the ratio of the distance from point $Q$ to the base $A_{2}$ of line segment $A_{2}B_{2}$ to the distance to the top $B_{2}$ is $1:2$, find the coordinates of point $Q$.

In a shooting game, a player must avoid obstacles to shoot a target. A rectangular coordinate system is set up on the game screen with the lower left corner $O$ of the screen as the origin, the lower edge of the screen as the $x$-axis, and the left edge of the screen as the $y$-axis. The target is placed at point $P ( 12,10 )$. There are two walls in the screen (wall thickness is negligible), one wall extends horizontally from point $A ( 10,5 )$ to point $B ( 15,5 )$, and another wall extends horizontally from point $C ( 0,6 )$ to point $D ( 9,6 )$, as shown in the schematic diagram on the right. If a player at point $Q$ can shoot the target at point $P$ in a straight line without being blocked by the two walls, which of the following options could be the coordinates of point $Q$?
(1) $( 6,3 )$
(2) $( 7,3 )$
(3) $( 8,5 )$
(4) $( 9,1 )$
(5) $( 9,2 )$

A resident of a building displays a Christmas tree-shaped light decoration on the building's exterior wall, as shown in the figure. From a certain point $P$ on the fifth floor exterior wall, small light bulbs are pulled to the two ends $A , B$ of the fourth floor to form an isosceles triangle $P A B$, where $\overline { P A } = \overline { P B }$; light bulbs are pulled to the two ends $C , D$ of the third floor to form an isosceles triangle $P C D$; light bulbs are pulled to the two ends $E , F$ of the second floor to form an isosceles triangle $PEF$. Assume each floor has equal height and each floor has equal length. If the length of the line segment cut out by the fifth floor inside triangle $P A B$ is $\frac { 1 } { 3 }$ of the floor length, what fraction of the floor length is the length of the line segment cut out by the fifth floor inside triangle $PEF$? (Light decoration thickness is negligible)
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 4 }$

On the coordinate plane, $P ( a , 0 )$ is a point on the $x$-axis, where $a > 0$. Let $L _ { 1 }$ and $L _ { 2 }$ be lines passing through point $P$ with slopes $- \frac { 4 } { 3 }$ and $- \frac { 3 } { 2 }$ respectively. Given that the difference in areas of the two right triangles formed by $L _ { 1 }$ and $L _ { 2 }$ with the two coordinate axes is 3, what is the value of $a$?
(1) $3 \sqrt { 2 }$
(2) 6
(3) $6 \sqrt { 2 }$
(4) 9
(5) $8 \sqrt { 2 }$

On the coordinate plane, given three points $A ( 0,2 )$ , $B ( - 1,0 )$ , $C ( 4,0 )$ . If the line $y = m x$ divides triangle $A B C$ into two equal areas, then $m = \frac { \text{(14--1)} } { \text{(14--2)} }$ . (Reduce to lowest terms)

ABCD is a rectangle $| \mathrm { AD } | = 1 \mathrm {~cm}$ $| \mathrm { AE } | = | \mathrm { EB } | = 2 \mathrm {~cm}$ $|FE| = \mathrm { x }$
According to the given information, what is x in cm?
A) $\frac { \sqrt { 3 } } { 2 }$
B) $\frac { \sqrt { 5 } } { 2 }$
C) $\frac { \sqrt { 3 } } { 3 }$
D) $\frac { \sqrt { 5 } } { 3 }$
E) $\frac { \sqrt { 7 } } { 3 }$

ABCD is a parallelogram AECD is a trapezoid $| \mathrm { BE } | = 3 \mathrm {~cm}$ $| \mathrm { DC } | = 4 \mathrm {~cm}$
If the area of the parallelogram ABCD in the figure is $20 \mathrm {~cm} ^ { 2 }$, what is the area of triangle $CBE$ in $\mathbf { cm } ^ { \mathbf { 2 } }$?
A) 7
B) 7,5
C) 8
D) 8,5
E) 9

In the Cartesian coordinate plane, the perpendicular drawn from point $A ( 1,0 )$ to the line $\mathbf { y } + \mathbf { 2 x } - \mathbf { 1 } = \mathbf { 0 }$ intersects the Y-axis at which point?
A) $\frac { - 1 } { 2 }$
B) $\frac { - 1 } { 3 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 1 } { 5 }$
E) $\frac { - 1 } { 6 }$

The vertices of a parallelogram with diagonals $[ AC ]$ and $[ BD ]$ are $\mathrm { A } ( 3,1 ) , \mathrm { B } ( 5,3 ) , \mathrm { C } ( 2,5 )$ and $\mathrm { D } ( \mathrm { a } , \mathrm { b } )$. What is the length of diagonal $[ BD ]$ in units?
A) 1
B) 2
C) 3
D) 4
E) 5

OABC is a parallelogram $\mathrm { A } = ( 5,0 )$ $\mathrm { C } = ( 3,4 )$
According to the given information above, what is the sum of the diagonal lengths of parallelogram OABC in units?
A) $5 \sqrt { 5 }$
B) $6 \sqrt { 5 }$
C) $7 \sqrt { 5 }$
D) $7 \sqrt { 3 }$
E) $8 \sqrt { 3 }$

In the right coordinate plane shown in the figure, lines d and e are perpendicular to each other.
Accordingly, what is the abscissa of the point where line d intersects the x-axis?
A) $\frac { 9 } { 2 }$
B) $\frac { 11 } { 2 }$
C) $\frac { 13 } { 3 }$
D) $\frac { 14 } { 3 }$
E) $\frac { 25 } { 6 }$

What is the perimeter of rectangle ABCD given in the coordinate plane in the figure?
A) 18
B) 21
C) 24
D) 27
E) 30

In the rectangular coordinate plane, the sides of rectangle $A B C D$ are parallel to the axes.
If the coordinates of vertices A and $C$ are $(1, -1)$ and $(3, 5)$ respectively, what is the area of rectangle ABCD in square units?
A) 8 B) 10 C) 12 D) 15 E) 16

According to the given information above, what is the sum of the coordinates of point C?
A) 3 B) 4 C) 5 D) 6 E) 7

In the rectangular coordinate plane, the line $y = \frac { x } { 7 }$ intersects the lines $x = 2$ and $x = 9$ at points $P$ and $R$ respectively.
Accordingly, what is the length $| \mathrm { PR } |$ in units?
A) $5 \sqrt { 2 }$
B) $6 \sqrt { 2 }$
C) $4 \sqrt { 10 }$
D) 8
E) 9

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