Orienting Yourself: The Use of Coordinates - Introduction

Identify the quadrant or axis in which the following points lie: $A(-3, 5)$, $B(4, -2)$, $C(0, -7)$, and $D(-2, -2)$.

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

For $A(-3, 5)$, $x < 0$ and $y > 0$ (Quadrant II). For $B(4, -2)$, $x > 0$ and $y < 0$ (Quadrant IV). For $C(0, -7)$, since the abscissa is $0$, it lies on the $y$-axis. For $D(-2, -2)$, both $x < 0$ and $y < 0$ (Quadrant III).

Write the coordinates of a point whose abscissa is $5$ and which lies on the $x$-axis.

$(5, 0)$

If a point lies on the $x$-axis, its ordinate ($y$-coordinate) must be $0$. Given the abscissa $x = 5$, the ordered pair is $(5, 0)$.

A point $P$ is at a distance of $3$ units from the $x$-axis and $4$ units from the $y$-axis. If it lies in the third quadrant, find its coordinates.

$P(-4, -3)$

The distance from the $y$-axis gives the absolute value of the $x$-coordinate, so $|x| = 4$. The distance from the $x$-axis gives the absolute value of the $y$-coordinate, so $|y| = 3$. Since the point lies in the third quadrant, both $x$ and $y$ must be negative. Thus, $x = -4$ and $y = -3$.

$A=(2, 3), B=(-2, 3), C=(-2, -3), D=(2, -3)$ By connecting $A \to B \to C \to D \to A$, we form a rectangle.

The points form a rectangle because the opposite sides are parallel and the adjacent sides are perpendicular. The length is $4$ units (from $-2$ to $2$) and the width is $6$ units (from $-3$ to $3$).

Point $P = (3, 4)$ Reflection in $x$-axis keeps the $x$-coordinate same and negates the $y$-coordinate. $Q = (3, -4)$

When a point is reflected across the $x$-axis, its distance from the $x$-axis remains the same, but it moves to the opposite side. Thus, the ordinate $4$ becomes $-4$, while the abscissa $3$ remains unchanged.