Representation on the Cartesian plane
On a map, in a video game or on a ship's radar screen, every position is given with two numbers. The first says how far to the right (or left); the second, how far up (or down). That pair of numbers and the axes that measure them are called the Cartesian plane.
The coordinate axes
Draw a set of coordinate axes in your notebook and identify which quadrant (first, second, third, or fourth) the following points are in without plotting them: $P(4, 7)$…
Draw a set of coordinate axes in your notebook and identify which quadrant (first, second, third, or fourth) the following points are in without plotting them:
-
$P(4, 7)$
-
$Q(-3, 5)$
-
$R(-2, -8)$
-
$S(6, -1)$
You can do it on paper and send a photo with your answer.
The coordinates of a point
Each point is written $P(x, y)$. To plot it: first count $x$ steps from $O$ to the right (or to the left if it is negative), then go up or down $y$ steps from there.
In the following Cartesian plane, the positions of four objects in a room are marked. Write down the coordinates $(x, y)$ of each of the points represented: the remote…
In the following Cartesian plane, the positions of four objects in a room are marked. Write down the coordinates $(x, y)$ of each of the points represented: the remote control ($A$), the book ($B$), the keys ($C$) and the backpack ($D$).
We are designing a level for a video game on a grid. Three of the vertices of a rectangular room are the points $A(1, 2)$, $B(6, 2)$ and $C(6, 5)$. a) Plot the points on…
We are designing a level for a video game on a grid. Three of the vertices of a rectangular room are the points $A(1, 2)$, $B(6, 2)$ and $C(6, 5)$.
- a) Plot the points on a Cartesian plane.
- b) What must the coordinates of the fourth vertex $D$ be so that the room is a perfect rectangle?
- c) Join the points in order ($A$, $B$, $C$ and $D$) and check your answer.
You can do it on paper and send a photo with your answer.
The quadrants
Depending on the sign of $x$ and of $y$, each point falls in one of the four quadrants:
A group of archaeologists is mapping the ruins of an ancient Roman city. They have placed the center of an amphitheater at the origin of coordinates $O(0,0)$. Plot the…
A group of archaeologists is mapping the ruins of an ancient Roman city. They have placed the center of an amphitheater at the origin of coordinates $O(0,0)$. Plot the following points of interest on a Cartesian plane (in your notebook, using grid paper) and determine which quadrant each one is in:
-
Temple of Jupiter: $A(-4, 3)$
-
Public baths: $B(5, -2)$
-
North Gate: $C(0, 6)$
-
Forum: $D(-3, -5)$
-
Market: $E(6, 4)$
If they decided to build a fence joining points $E$, $B$ and $A$, what type of geometric figure would be formed?
Look at the points represented on the following marine navigation radar. Write down the coordinates of the ships identified by the letters $P$, $Q$, $R$ and $S$. Also…
Look at the points represented on the following marine navigation radar. Write down the coordinates of the ships identified by the letters $P$, $Q$, $R$ and $S$. Also indicate which ships have the same x-coordinate and which ones lie on one of the coordinate axes.
Points on the axes and symmetrical points
A point can lie on an axis instead of in a quadrant: if $x = 0$ it is on the y-axis; if $y = 0$, it is on the x-axis. The origin $(0, 0)$ is not in any quadrant.
It is also useful to know how to reflect a point:
In a strategy video game, the base of operations is at point $M(-3, 2)$. The programmer wants to place three guard posts in symmetrical positions: Post $P_1$ is the…
In a strategy video game, the base of operations is at point $M(-3, 2)$. The programmer wants to place three guard posts in symmetrical positions:
-
Post $P_1$ is the reflection of $M$ across the x-axis.
-
Post $P_2$ is the reflection of $M$ across the y-axis.
-
Post $P_3$ is the reflection of $M$ in the origin.
Plot the four points on the Cartesian plane and write down their coordinates.
Complete the following sentences about the Cartesian coordinate system: a) The horizontal axis is called the axis of \\\\\\\\ and is represented by the letter $X$. b)…
Complete the following sentences about the Cartesian coordinate system:
- a) The horizontal axis is called the axis of ________ and is represented by the letter $X$.
- b) The vertical axis is called the axis of ________ and is represented by the letter $Y$.
- c) The point of intersection of both axes is called the ________ and its coordinates are $(0, 0)$.