Orienting Yourself: The Use of Coordinates - The 2-d Cartesian Coordinate System

In which quadrant or on which axis do the following points lie? $A(4, -3)$, $B(-2, -5)$, $C(0, 5)$, and $D(-3, 0)$.

$A$ lies in Quadrant IV, $B$ lies in Quadrant III, $C$ lies on the $y$-axis, and $D$ lies on the $x$-axis.

For $A(4, -3)$, $x > 0$ and $y < 0$, which corresponds to Quadrant IV. For $B(-2, -5)$, $x < 0$ and $y < 0$, which is Quadrant III. For $C(0, 5)$, since the $x$-coordinate is $0$, it lies on the $y$-axis. For $D(-3, 0)$, since the $y$-coordinate is $0$, it lies on the $x$-axis.

Write the coordinates of a point $P$ whose ordinate is $3$ and abscissa is $-4$.

$P(-4, 3)$

The abscissa refers to the $x$-coordinate and the ordinate refers to the $y$-coordinate. Writing them in the form $(x, y)$ gives $(-4, 3)$.

If the coordinates of a point are $P(2, 3)$, find its perpendicular distance from the $x$-axis and the $y$-axis.

Distance from $x$-axis = $3$ units; Distance from $y$-axis = $2$ units.

The distance from the $x$-axis is given by the absolute value of the $y$-coordinate (ordinate), which is $|3| = 3$. The distance from the $y$-axis is given by the absolute value of the $x$-coordinate (abscissa), which is $|2| = 2$.

$A(2, 3) \in \text{Quadrant I}$ (both positive) $B(-3, 2) \in \text{Quadrant II}$ ($x$ negative, $y$ positive) $C(-2, -2) \in \text{Quadrant III}$ (both negative) $D(4, -1) \in \text{Quadrant IV}$ ($x$ positive, $y$ negative)

By checking the signs of the $x$ and $y$ coordinates, we determine the quadrant. Point $A$ is in the top-right, $B$ is top-left, $C$ is bottom-left, and $D$ is bottom-right.

The vertices of the rectangle are $O(0, 0)$, $A(5, 0)$ on the $x$-axis, $M(5, 3)$, and $B(0, 3)$ on the $y$-axis. Length $OA = |5 - 0| = 5$ units. Width $OB = |3 - 0| = 3$ units. $\text{Area} = \text{Length} \times \text{Width}$ $\text{Area} = 5 \times 3 = 15 \text{ sq. units}$

The coordinates $(5, 3)$ define a rectangle with the axes where the length is the abscissa and the height is the ordinate.