QUESTION 179
The midpoint of the segment with endpoints $A(2, 4)$ and $B(6, 8)$ is
(A) $(3, 5)$
(B) $(4, 6)$
(C) $(5, 7)$
(D) $(6, 8)$
(E) $(8, 12)$

Consider a triangle $ABC$. The sides $AB$ and $AC$ are extended to points $D$ and $E$, respectively, such that $AD = 3AB$ and $AE = 3AC$. Then one diagonal of $BDEC$ divides the other diagonal in the ratio
(A) $1 : 3$
(B) $1 : \sqrt{3}$
(C) $1 : 2$
(D) $1 : \sqrt{2}$.

The lines $L_{1}: y-x=0$ and $L_{2}: 2x+y=0$ intersect the line $L_{3}: y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

A straight line through origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio: (1) $2:3$ (2) $1:2$ (3) $4:1$ (4) $3:4$

A straight line through origin $O$ meets the lines $3 y = 10 - 4 x$ and $8 x + 6 y + 5 = 0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $A B$ in the ratio
(1) $2 : 3$
(2) $1 : 2$
(3) $4 : 1$
(4) $3 : 4$

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").
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 $Q$.

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

The square ABCD given above is divided into four rectangles of equal area.
Accordingly, what is the ratio $\frac { | AE | } { | AD | }$?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 3 } { 5 }$\ D) $\frac { 5 } { 8 }$\ E) $\frac { 9 } { 16 }$