Cartesian System Questions in English

(C) In a Cartesian coordinate system,the origin is the point where the $x$-axis and $y$-axis intersect. By definition,the coordinates of the origin are $(0, 0)$.

(B) In the Cartesian coordinate system,the horizontal line is known as the $x$-axis and the vertical line passing through the origin $(0, 0)$ is known as the $y$-axis. Therefore,the correct answer is the $y$-axis.

(C) In a Cartesian plane,the axes $XOX'$ and $YOY'$ divide the plane into four quadrants.
- The first quadrant is formed by the region between the positive $X$-axis and positive $Y$-axis.
- The second quadrant is formed by the region between the negative $X$-axis $(OX')$ and positive $Y$-axis $(OY)$.
- The third quadrant is formed by the region between the negative $X$-axis $(OX')$ and negative $Y$-axis $(OY')$.
- The fourth quadrant is formed by the region between the positive $X$-axis $(OX)$ and negative $Y$-axis $(OY')$.
Therefore,the interior of $\angle X'OY'$ corresponds to the third quadrant.

(B) In a Cartesian coordinate system,the quadrants are defined as follows:
$I$ quadrant: $(+, +)$
$II$ quadrant: $(-, +)$
$III$ quadrant: $(-, -)$
$IV$ quadrant: $(+, -)$
Given the point $\left(-\frac{7}{2}, \frac{5}{2}\right)$,the $x$-coordinate is negative and the $y$-coordinate is positive.
Therefore,the point lies in the $II$ quadrant.

(C) In a Cartesian plane,the $X$-axis and $Y$-axis divide the plane into four quadrants. The $X^{\prime}OX$ line represents the $X$-axis,where $OX$ is the positive $X$-axis and $OX^{\prime}$ is the negative $X$-axis. Similarly,$YOY^{\prime}$ represents the $Y$-axis,where $OY$ is the positive $Y$-axis and $OY^{\prime}$ is the negative $Y$-axis. The third quadrant is bounded by the negative $X$-axis $(OX^{\prime})$ and the negative $Y$-axis $(OY^{\prime})$. Therefore,the interior of $\angle X^{\prime}OY^{\prime}$ represents the third quadrant.

(A) In a Cartesian coordinate system,any point on the $y$-axis has a horizontal distance of $0$ from the origin. Therefore,the $x$-coordinate (also known as the abscissa) of any point on the $y$-axis is always $0$. The coordinates of a point on the $y$-axis are represented as $(0, y)$.

(B) In the Cartesian plane,a point $(x, y)$ has coordinates with opposite signs if one coordinate is positive and the other is negative.
Case $1$: $x > 0$ and $y < 0$. This corresponds to the Fourth quadrant.
Case $2$: $x < 0$ and $y > 0$. This corresponds to the Second quadrant.
Therefore,a point with coordinates of opposite signs lies in either the Second quadrant or the Fourth quadrant.

(A) The coordinates of the two points are $(8, 8)$ and $(-8, 8)$.
Since the $y$-coordinate is the same for both points $(y = 8)$,the line segment joining these points is a horizontal line.
$A$ horizontal line is always parallel to the $x$-axis.
Therefore,the line is parallel to the $x$-axis.

(D) In a Cartesian coordinate system,the $X$-axis and $Y$-axis divide the plane into four quadrants.
- The first quadrant is bounded by ray $OX$ and ray $OY$.
- The second quadrant is bounded by ray $OX'$ and ray $OY$.
- The third quadrant is bounded by ray $OX'$ and ray $OY'$.
- The fourth quadrant is bounded by ray $OX$ and ray $OY'$.
Therefore,the region bounded by ray $OY'$ and ray $OX$ is the fourth quadrant.

(D) In a Cartesian coordinate system,the coordinate axes consist of the $x$-axis and the $y$-axis.
These two axes intersect each other at the origin $(0, 0)$.
By definition,the $x$-axis and the $y$-axis are perpendicular to each other.
Therefore,the angle formed by the intersection of the coordinate axes is $90^{\circ}$.

(B) In a Cartesian coordinate system,the ray $OX$ represents the positive $x$-axis,where $x > 0$. The ray $OX^{\prime}$ represents the negative $x$-axis,where $x < 0$. Since the point $(a, 0)$ lies on the ray $OX^{\prime}$,the $x$-coordinate $a$ must be a negative number.

(A) point is said to lie on the $x$-axis if its $y$-coordinate is $0$.
Given the point $\left(7 \frac{3}{4}, 0\right)$,the $y$-coordinate is $0$.
Therefore,the point lies on the $x$-axis.

(B) In a Cartesian plane,the quadrants are defined as follows:
$1$. First quadrant: $x > 0, y > 0$
$2$. Second quadrant: $x < 0, y > 0$
$3$. Third quadrant: $x < 0, y < 0$
$4$. Fourth quadrant: $x > 0, y < 0$
Since the given point has a negative $x$-coordinate and a positive $y$-coordinate,it lies in the second quadrant.

(C) Given values are $a=5, b=3, c=-8, d=-5$.
We need to find the coordinates of the point $(a+c, b+d)$.
First,calculate the $x$-coordinate: $a+c = 5 + (-8) = -3$.
Next,calculate the $y$-coordinate: $b+d = 3 + (-5) = -2$.
The point is $(-3, -2)$.
Since both the $x$-coordinate and $y$-coordinate are negative,the point lies in the third quadrant.

(B) In a Cartesian plane,the equation $x=0$ represents all points where the $x$-coordinate is $0$. Since the $x$-coordinate is $0$ for every point on the $y$-axis,the graph of $x=0$ is the $y$-axis.