Representation in the Plane

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 they intersect.

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

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:

1. The fountain ($A$)

2. The bench ($B$)

3. The trash can ($C$)

4. 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:

• $A(2, 3)$

• $B(-2, 3)$

• $C(-2, -3)$

• $D(2, -3)$

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.

In a strategy video game, the main base is located at the origin of coordinates $O(0, 0)$. The radar detects three enemy targets at 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.

You can do it on paper and send a photo with your answer.

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: $P_1(5, -2)$

• 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:

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:

• $A(0, -5)$

• $B(-2, -8)$

• $C(4, 0)$

• $D(-1, 7)$

• $E(3, -3)$

Look at the following figure, which shows the line segment connecting the points $A(-4, 2)$ and $B(2, -1)$.

1. State the coordinates of the point where the segment crosses the x-axis ($X$ axis).

2. State the coordinates of the point where the segment crosses the y-axis ($Y$ axis). What is special about both points of intersection?

3. Which of these expressions relates the coordinates of the points on the segment: $y = 2x$, $y = -\frac{x}{2}$ or $y = x - 2$? Verify your choice by substituting the coordinates of $A$ and $B$.

Imagine that you are designing a treasure map. The starting point is the origin of coordinates $O(0,0)$ and each grid square of the plane represents 1 unit. The instructions say:

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

1. a) Write the coordinates of each point you pass through at the end of each leg of the route.

1. b) State the coordinates of the final point where the treasure is located.

1. c) Copy the plane in your notebook and plot it: draw the complete route and mark the treasure point.

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

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

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

1. a) Plot 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?

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

1. a) Plot the resulting points on a Cartesian coordinate system.
2. b) What geometric shape do the points appear 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.

You can do it on paper and send a photo with your answer.

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

1. a) Represent both points on the Cartesian plane.

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

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

1. a) Plot 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?

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 use graph paper) and draw triangle $ABC$.

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

3. Identify the height corresponding to that base and how long it is (units on the $Y$ axis).

4. 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 $R(4, 3)$.

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

A surveillance drone flies over an industrial area represented on a Cartesian plane. The drone starts 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 route, where the drone must stop to take a high-resolution photograph.

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

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

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

1. a) Plot the points on the Cartesian coordinate system and determine the coordinates of the fourth vertex $D$ so that the points, taken in alphabetical order $A$, $B$, $C$, $D$, form a parallelogram.

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

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

You can do it on paper and send a photo with your answer.

Consider a triangle whose vertices are $A(2, 1)$, $B(6, 1)$ and a third, variable vertex $C$. In the figure you have the points $A$ and $B$ already placed on the coordinate axes.

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 placed at the point $(a, b)$ in such a way that triangle $ABC$ has a right angle at $A$, has the same area as in the previous part ($10$ square units) and $C$ is also in the first quadrant, what are the coordinates of $C$?

1. c) Graph both triangles on the same coordinate axes.

You can do it on paper and send a photo with your answer.

In the design of a circular logo, the center of the circle is at point $C(1, 2)$. It is known that point $P(4, 6)$ lies on the edge of the logo.

1. a) Plot 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 line segment $PQ$ represent in the circle?

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 of point $P'$ be?

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 plotting these points and joining them in order.

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

1. c) If we double the $y$ coordinate of all vertices while keeping the $x$ coordinate unchanged, how does the area of the resulting figure change compared to the original?

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

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

2. Its y-coordinate (ordinate) is the modulus of the solution to $\frac{y}{2} - 5 = -8$.

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

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

1. b) Check whether the point $S$ found satisfies condition 3. Justify your answer by plotting the direct proportion function.

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

A rescue boat is at point $R(0, 5)$ and receives a call from a castaway at point $N(8, -1)$. Because of the currents, the boat can only sail along horizontal and vertical stretches (following the coordinate axes and lines parallel to them) or else moving in a straight line.

1. a) Plot both points on a coordinate system and draw the shortest path using horizontal and vertical stretches. 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 with the previous path?

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

Consider the geometric figure 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 coordinate $a$ (considering $a > 0$).

1. b) Plot the triangle on the plane and draw its reflection across 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 check whether the distance from $M$ to the origin $O$ is the same as the distance from $M$ to point $A$.

You can do it on paper and send a photo with your answer.