Representation in the Plane
On a map on your phone, in a spreadsheet, or on a "Battleship" board, every square is located with two pieces of information: column and row. The Cartesian plane does the same thing with two numbers, and it is the stage where all functions are drawn.
Axes and Coordinates
Drag the point or tap anywhere on the plane:
In the map app on your cell phone, your local park is shown on a grid using a Cartesian coordinate system. Look at the position of the different elements and write down…
In the map app on your cell phone, your local park is shown on a grid using a Cartesian coordinate system. Look at the position of the different elements and write down the coordinates of the points where they are located:
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The fountain ($A$)
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The bench ($B$)
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The trash can ($C$)
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The lamppost ($D$)
Draw a Cartesian coordinate system in your notebook and plot the following points, which form the vertices of a safe landing zone: $A(2, 3)$ $B(-2, 3)$ $C(-2, -3)$…
Draw a Cartesian coordinate system in your notebook and plot the following points, which form the vertices of a safe landing zone:
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$A(2, 3)$
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$B(-2, 3)$
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$C(-2, -3)$
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$D(2, -3)$
What geometric figure do you get when you join the points in order?
You can do it on paper and send a photo with your answer.
Identify the name of the elements indicated on this Cartesian reference system: 1. The horizontal line (... axis) 2. The vertical line (... axis) 3. Point $(0,0)$ where…
Identify the name of the elements indicated on this Cartesian reference system:
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The horizontal line (... axis)
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The vertical line (... axis)
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Point $(0,0)$ where they intersect.
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The four regions into which the plane is divided.
Quadrants and Points on the Axes
The axes divide the plane into four quadrants, numbered counterclockwise. The signs of the coordinates tell you which quadrant a point is in, without drawing it:
Without plotting them, state which quadrant the following points lie in, which represent the position of several players on a video game server: Player 1: $P_1(5, -2)$…
Without plotting them, state which quadrant the following points lie in, which represent the position of several players on a video game server:
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Player 1: $P_1(5, -2)$
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Player 2: $P_2(-3, -8)$
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Player 3: $P_3(-1, 4)$
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Player 4: $P_4(2, 6)$
State the coordinates of the following special points located on the coordinate axes and explain why their position is characteristic:
State the coordinates of the following special points located on the coordinate axes and explain why their position is characteristic:
Without plotting the points, state which quadrant or axis the following points lie on based on the signs of their coordinates. Justify your answer by explaining whether…
Without plotting the points, state which quadrant or axis the following points lie on based on the signs of their coordinates. Justify your answer by explaining whether the x-coordinate (abscissa) or y-coordinate (ordinate) is positive, negative or zero:
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$A(0, -5)$
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$B(-2, -8)$
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$C(4, 0)$
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$D(-1, 7)$
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$E(3, -3)$
Calculating with Coordinates
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Horizontal distance between two points with the same $y$: subtract the $x$-coordinates (in absolute value). For $A(3, 2)$ and $B(-1, 2)$: $|3 - (-1)| = 4$.
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Vertical distance between two points with the same $x$: subtract the $y$-coordinates.
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Any distance: the Pythagorean theorem with the two differences: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
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Midpoint of a segment: the average of the coordinates, $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
Three vertices of a rectangle are $A(-3, 1)$, $B(4, 1)$ and $C(4, -2)$. a) Plot the three points on a Cartesian plane. b) Find the coordinates of the fourth vertex $D$…
Three vertices of a rectangle are $A(-3, 1)$, $B(4, 1)$ and $C(4, -2)$.
- a) Plot the three points on a Cartesian plane.
- b) Find the coordinates of the fourth vertex $D$ to complete the rectangle.
- c) In which quadrant is the new point $D$ located?
Calculate the area of a triangle whose vertices are the points $A(-2, 1)$, $B(4, 1)$ and $C(1, 6)$. To solve it: 1. Plot the points on a Cartesian plane (it will help to…
Calculate the area of a triangle whose vertices are the points $A(-2, 1)$, $B(4, 1)$ and $C(1, 6)$.
To solve it:
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Plot the points on a Cartesian plane (it will help to use graph paper) and draw triangle $ABC$.
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Identify which side is the base of the triangle and how long it is (units on the $X$ axis).
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Identify the height corresponding to that base and how long it is (units on the $Y$ axis).
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Apply the formula for the area of a triangle: $Área = \frac{base \times altura}{2}$.
Write the result in square units.
Three of the vertices of a rectangle intended for a skating area are located at points $P(-3, -2)$, $Q(4, -2)$ and $R(4, 3)$. a) Plot the points on the plane and draw…
Three of the vertices of a rectangle intended for a skating area are located at points $P(-3, -2)$, $Q(4, -2)$ and $R(4, 3)$.
- a) Plot the points on the plane and draw the rectangle.
- b) Determine the coordinates of the fourth vertex $S$ so that the figure is a rectangle.
- c) Calculate the perimeter of the skating area knowing that each unit on the plane is equivalent to $2$ meters.