The Coordinate Plane

Identify the name of the elements indicated in this Cartesian reference system:

1. The horizontal line (... axis)

2. The vertical line (... axis)

3. The point $(0,0)$ where they cross.

4. The four regions into which the plane is divided.

A drone is flying over a park that we have gridded on a Cartesian coordinate system. Observe the position of the different elements and write the coordinates of the points where they are located:

1. The fountain ($A$)

2. The bench ($B$)

3. The trash can ($C$)

4. The lamppost ($D$)

Imagine you are designing a treasure map. The starting point is the origin of coordinates $O(0,0)$. The instructions say:

"Walk 4 units to the East (right), then 3 units to the North (up), then 6 units to the West (left) and finally 2 units to the South (down)".

Write the coordinates of the final point where the treasure is located and represent it graphically.

Without needing to represent them graphically, state in which quadrant the following points, which indicate the position of several players on a video game server, are located:

• Player 1: $P_1(5, -2)$

• Player 2: $P_2(-3, -8)$

• Player 3: $P_3(-1, 4)$

• 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:

(Consider that each mark on the axis represents 3 units).

Without needing to represent the points, state in which quadrant or axis the following points are located based on the sign of their coordinates. Justify your answer by reasoning whether the x-coordinate or y-coordinate are positive, negative or zero:

• $A(0, -5)$

• $B(-2, -8)$

• $C(4, 0)$

• $D(-1, 7)$

• $E(3, -3)$

On a social network, the position of two friends is given by points $A(3, 2)$ and $B(3, -4)$.

1. a) Represent them on the plane.
2. b) What do their coordinates have in common?
3. c) If we draw a line passing through both, would it be a horizontal or vertical line?

In a strategy video game, the main base is located at the origin of coordinates $O(0, 0)$. The radar detects three enemy targets in the following positions: target $A$ at $(-4, 3)$, target $B$ at $(2, 5)$ and target $C$ at $(3, -2)$.

Represent the points on the Cartesian plane and state in which quadrant each of the enemy targets is located.

Draw a Cartesian coordinate system in your notebook and represent the following points, which form the vertices of a safe landing zone:

• $A(2, 3)$

• $B(-2, 3)$

• $C(-2, -3)$

• $D(2, -3)$

What geometric shape do you obtain when joining the points in order?

Three vertices of a rectangle are $A(-3, 1)$, $B(4, 1)$ and $C(4, -2)$.

1. a) Represent the three points on a Cartesian plane.
2. b) Find the coordinates of the fourth vertex $D$ to complete the rectangle.
3. c) In which quadrant is the new point $D$ located?

Three of the vertices of a rectangle intended for a skating area are located at points $P(-3, -2)$, $Q(4, -2)$ and $R(4, 3)$.

1. a) Represent the points on the plane and draw the rectangle.
2. b) Determine the coordinates of the fourth vertex $S$ so that the shape is a rectangle.
3. c) Calculate the perimeter of the skating area knowing that each unit on the plane is equivalent to $2$ feet.

The logo of a new clothing brand is symmetrical with respect to the y-axis ($Eje\ Y$). The right part of the logo is defined by points $A(0, 5)$, $B(2, 4)$ and $C(3, 0)$.

1. a) Obtain the coordinates of the symmetrical points $B'$ and $C'$ with respect to the $Y$ axis.
2. b) Represent the five points on the plane and join them in order ($C'-B'-A-B-C$) to visualise the shape of the logo.

In the design of a circular logo, the center of the circumference is located at point $C(1, 2)$. It is known that point $P(4, 6)$ belongs to the edge of the logo.

1. a) Represent the center and point $P$ on the plane.

1. b) Find another point $Q$ that is symmetrical to point $P$ with respect to the center $C$. What does the segment $PQ$ represent in the circumference?

1. c) If the logo is translated 4 units to the left and 2 units down, what will the new coordinates of the center $C'$ and point $P'$ be?

A surveillance drone makes a journey over an industrial area represented on a Cartesian plane. The drone sets off from point $P(-6, -4)$ and flies in a straight line to point $Q(2, 6)$.

1. a) Determine the coordinates of the midpoint of the journey, where the drone must stop to take a high-resolution photograph.

1. b) If the drone deviates from its original route and moves from the midpoint calculated in the previous section 5 units to the right and 3 units down, in which quadrant will it be?

1. c) Calculate the total distance the drone would travel if it went directly from the origin of coordinates $O(0,0)$ to point $Q(2,6)$.

A rescue boat is located at point $R(0, 5)$ and receives a warning of a castaway at point $N(8, -1)$. Due to the currents, the boat can only sail following the edges of the quadrants (first along the $Y$ axis and then along the $X$ axis, or vice versa) or by moving in a straight line.

1. a) Represent both points and draw the shortest path following the axes. What total distance would it travel if each unit on the plane is 10 nautical miles?

1. b) Calculate the straight-line (diagonal) distance between the boat and the castaway. How many nautical miles would be saved compared to the previous path?

1. c) If a second castaway is located at the point symmetrical to $N$ with respect to the $Y$ axis, what are their coordinates?

Given the polygon with vertices $A(-2, 2)$, $B(2, 2)$, $C(4, -1)$ and $D(-4, -1)$:

1. a) Classify the quadrilateral obtained by representing these points and joining them in order.

1. b) Calculate its approximate perimeter. (Hint: use Pythagoras' theorem to calculate the length of the oblique sides).

1. c) If we double the $y$ coordinate of all the vertices but keep the $x$ coordinate intact, how does the area of the resulting shape change with respect to the original?

Calculate the area of a triangle whose vertices are points $A(-2, 1)$, $B(4, 1)$ and $C(1, 6)$.

To solve it:

1. Represent the points on the plane.

2. Identify which is the base of the triangle and how long it is (units of the $X$ axis).

3. Identify the height and how long it is (units of the $Y$ axis).

4. Apply the formula for the area of a triangle: $Área = \frac{base \cdot altura}{2}$.

Consider a triangle whose vertices are $A(2, 1)$, $B(6, 1)$ and a third variable vertex $C$.

1. a) If vertex $C$ has coordinates $(4, y)$, calculate the value of $y$ so that the area of the triangle is $10$ square units, knowing that $C$ is in the first quadrant.

1. b) If vertex $C$ is now located at point $(a, b)$ such that triangle $ABC$ is a right triangle at $A$ and has the same area as in the previous section, what are the coordinates of $C$?

1. c) Represent both triangles graphically on the same coordinate system.

Three vertices of a parallelogram are the points $A(-4, 1)$, $B(2, 1)$ and $C(4, 5)$. Use your knowledge of representation on the plane to carry out the following tasks:

1. a) Represent the points on the Cartesian coordinate system and determine the coordinates of the fourth vertex $D$ so that the points, followed in alphabetical order, form the shape.

1. b) Calculate the area of the parallelogram taking the grid square as the unit.

1. c) If we reflect point $A$ across the origin of coordinates $(0,0)$, in which quadrant would the new point $A'$ be located and what would its coordinates be?

Let the geometric shape be defined by the points $O(0,0)$, $A(a, 0)$ and $B(0, a)$.

1. a) If the area of triangle $OAB$ is $18$ square units, determine the value of the $a$ coordinate (considering $a > 0$).

1. b) Represent the triangle on the plane and draw its symmetrical shape with respect to the angle bisector of the first quadrant (the line where $x = y$). What do you observe about the coordinates of the new points?

1. c) Calculate the coordinates of point $M$, the midpoint of segment $AB$, and verify if the distance from $M$ to the origin $O$ is the same as the distance from $M$ to point $A$.

Given the following table of values that relates the input ($x$) to the output ($y$) of a function:

1. a) Represent the resulting points on a Cartesian coordinate system.
2. b) What geometric shape do the points seem to form if we were to join them with a smooth curved line?
3. c) State the coordinates of the point where the function reaches its minimum value.

A surveillance drone makes a photographic route following a straight-line trajectory. It is known that it passes through the control point $M(1, 2)$ and the supply point $N(3, 6)$.

1. a) Represent both points on the Cartesian plane.
2. b) If the drone continues in the same direction, state whether it will pass through point $P(5, 10)$ or point $Q(4, 7)$. Check this graphically by drawing the line that joins $M$ and $N$.

Determine the coordinates of the points of intersection with the axes of the geometric shape formed by joining points $A(-4, 2)$ and $B(2, -1)$ with a line segment.

1. Represent points $A$ and $B$.

2. Draw the segment that joins them.

3. Observe the graph and state the exact point where the segment crosses the x-axis ($X$) and the y-axis ($Y$).

We are looking for a secret point $S(x, y)$ that satisfies the following logical conditions:

1. Its x-coordinate is the integer that solves the equation $3(x - 2) = x + 4$.

2. Its y-coordinate is the absolute value of the solution to $\frac{y}{2} - 5 = -8$.

3. The point belongs to a direct proportion function that passes through point $(1, -2)$.

1. a) Find the coordinates of point $S$ by solving conditions 1 and 2.

1. b) Check if the point $S$ found satisfies condition 3. Justify your answer by graphically representing the proportion function.

1. c) State in which quadrant point $S$ is located.