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Lesson Explainer: Coordinate Planes Mathematics

Exercise 1

Consider the number line below and the point 𝐹 midpoint of line segment 𝐴𝐡.

  1. What is the π‘₯-coordinate of each of the points 𝐴, 𝐡, 𝐢, 𝐷, and 𝐹?
  2. Read the π‘₯-coordinate of the point symmetric to point 𝐷 with respect to the point 𝐢.

Exercise 2

Consider the orthonormal coordinate plane (𝑂;𝐼, 𝐽).

  1. Which of the following points 𝐴, 𝐡, 𝐢, 𝐷, and 𝐹 have:
    1. The same π‘₯-coordinate?
    2. The same 𝑦-coordinate?
  2. Give the coordinates of the points 𝐴, 𝐡, 𝐢, 𝐷, and 𝐹.
  3. Is the following statement true or false?
    β€œIn the above figure, there are as many points that have a positive 𝑦-coordinate as there are points that have a negative π‘₯-coordinate.”

Exercise 3

Consider a triangle 𝐴𝐡𝐢 right-angled at 𝐡 such that:

𝐴𝐡=7cm and 𝐴𝐢=10cm.

1. What is the exact length of line segment 𝐡𝐢?

Consider a triangle 𝐷𝐸𝐹 such that:

𝐷𝐸=4cm; 𝐷𝐹=9.6cm and 𝐸𝐹=10.4cm.

2. Is this triangle right-angled?

Activity 1: Grid

  1. Plot on a grid the points 𝐴, 𝐡, and 𝐢 as shown below.
  2. Plot the points 𝑂, 𝐼, and 𝐽 such that the point 𝐴 has coordinates (3,1) in the coordinate plane (𝑂;𝐼, 𝐽).
  3. What are the coordinates of points 𝐡 and 𝐢 in this coordinate plane?

Activity 2: Coordinates of midpoint

  1. In an orthonormal coordinate plane (𝑂;𝐼, 𝐽), plot the points 𝐸(2,3), 𝐹(6,1), and 𝐺(1,βˆ’3).
  2. Read the coordinates of points 𝑅 and 𝐿 the midpoints of line segments 𝐸𝐹 and 𝐹𝐺 respectively.
  3. What is the relation that exists between the coordinates of point 𝑅 and the coordinates of points 𝐸 and 𝐹?
  4. Does this relation remain valid for point 𝐿?

Activity 3: Length of a line segment

  1. In an orthonormal coordinate plane (𝑂;𝐼, 𝐽) with 1 cm of unit length, place the points 𝐴(3,1) and 𝐡(7,4).
  2. To find the distance 𝐴𝐡 in this coordinate plane:
    1. Plot the point 𝐾(7,1). What is the type of triangle 𝐴𝐾𝐡?
    2. Find the distances 𝐴𝐾 and 𝐡𝐾. Deduce the distance 𝐴𝐡.
  3. Plot the point 𝐢(2,6).
    Calculate the distance 𝐴𝐢 using the method used in question 2.

I. Coordinates of a point in a plane

(1) Coordinate plane

Definition

A coordinate plane is formed by any triplet of non collinear points (𝑂;𝐼, 𝐽).

Point 𝑂 is the origin of the coordinate plane, the line ⃖⃗𝑂𝐼 is the π‘₯-axis and the line ⃖⃗𝑂𝐼 is the 𝑦-axis.

The length of line segment 𝑂𝐼 is the unit length of the π‘₯-axis and the length of the line segment 𝑂𝐽 is the unit length of the 𝑦-axis.

Different types of coordinate planes

  • If the axes ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are intersecting at 𝑂 and are non-perpendicular, then (𝑂;𝐼, 𝐽) is called an oblique coordinate plane.
  • If the axes ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are perpendicular at 𝑂 and if 𝑂𝐼≠𝑂𝐽, then (𝑂;𝐼, 𝐽) is called an orthogonal coordinate plane.
  • If the axes ⃖⃗𝑂𝐼 and ⃖⃗𝑂𝐽 are perpendicular at 𝑂 and if, in addition, 𝑂𝐼=𝑂𝐽, then (𝑂;𝐼, 𝐽) is called an orthonormal coordinate plane.

Example 1: Finding a way on a grid of a chess game

Statement

The squares of the chess game are marked according to the code used in the below figure.

A knight (represented by a horse’s head) moves following a horizontal or vertical movement of one or two squares in a given direction, then two or one square following the perpendicular direction forming an L-shape.

A king is β€œchecked” by an opponent’s piece if he is found on a square that can be reached by this piece.

  1. Locate the squares containing the pieces of this chess game.
  2. A player wants to move the white knight located at d8. Identify the possible squares that can contain this knight.
  3. The rook moves vertically or horizontally as many squares as needed. Where to put the rook to β€œcheck” the black king?

Answer

  1. This chess game contains 5 pieces located as follows:
    • The white knight is found at d8,
    • The white rook is found at h7,
    • The black king is found at d3,
    • The white king is found at f3,
    • The black pawn is found at c2.
  2. The white knight located at d8 can reach one of the following squares: c6, e6, b7, and f7.
  3. In order to β€œcheck” the black king, the white rook has to move to d7. In fact, at d3, the white king is an obstacle.

(2) Locating a point in a coordinate plane

We start by defining a coordinate plane:

  • We choose the triplet (𝑂;𝐼, 𝐽) of non collinear points.

For a point 𝑀 in the plane:

  • π‘₯ is the π‘₯-coordinate on the axis ⃖⃗𝑂𝐼 of the point of intersection of axis ⃖⃗𝑂𝐼 with the line parallel to axis ⃖⃗𝑂𝐽 going through 𝑀.
  • π‘¦οŒ¬ is the π‘₯-coordinate on the axis ⃖⃗𝑂𝐽 of the point of intersection of axis ⃖⃗𝑂𝐽 with the line parallel to axis ⃖⃗𝑂𝐼 going through 𝑀.

In the coordinate plane (𝑂;𝐼, 𝐽), a point 𝑀 is plotted using the ordered pair (π‘₯,𝑦). The real numbers π‘₯ and π‘¦οŒ¬ are called the coordinates of point 𝑀 in the coordinate plane (𝑂;𝐼, 𝐽).

Notation

We write 𝑀(π‘₯,𝑦) to say that the real number π‘₯ is the π‘₯-coordinate of point 𝑀 and the real number π‘¦οŒ¬ is the 𝑦-coordinate of point 𝑀.

Example

  • The coordinate plane (𝑂;𝐼, 𝐽) in the figure is orthonormal.
  • The point 𝑀 has coordinates 𝑀(βˆ’1.5,5).
  • The origin 𝑂 has coordinates 𝑂(0,0).
  • The point 𝐼 has coordinates 𝐼(1,0).
  • The point 𝐽 has coordinates 𝐽(0,1).

II. Coordinates of midpoint of a line segment

Property

Let (𝑂;𝐼, 𝐽) a coordinate plane, 𝐴 and 𝐡 two points with coordinates (π‘₯,𝑦) and (π‘₯,𝑦) respectively.

The midpoint 𝑀 of line segment 𝐴𝐡 has coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

Example

Given the points with coordinates, 𝐴(βˆ’2,2) and 𝐡(4,0).

The midpoint of line segment 𝐴𝐡 has coordinates ο€Ό4βˆ’22;2+02 that is (1,1).

Proof

Consider the two points 𝐴(π‘₯;𝑦) and 𝐡(π‘₯;𝑦) in an orthonormal coordinate plane (𝑂;𝐼, 𝐽).

𝑀 is the midpoint of line segment 𝐴𝐡.

1st case:

Points 𝐴, 𝑀 and 𝐡 have the same 𝑦-coordinate.

We have 𝑀𝐴=𝑀𝐡 and 𝑦=𝑦=π‘¦οŒ¬οŒ οŒ‘.

So, π‘₯βˆ’π‘₯=π‘₯βˆ’π‘₯𝑦=𝑦=𝑦2π‘₯=π‘₯+π‘₯2𝑦=𝑦+𝑦π‘₯=π‘₯+π‘₯2𝑦=𝑦+𝑦2.andandand

2nd case:

Points 𝐴, 𝑀 and 𝐡 have the same π‘₯-coordinate.

We have 𝑀𝐴=𝑀𝐡 and π‘₯=π‘₯=π‘₯.

So,π‘¦βˆ’π‘¦=π‘¦βˆ’π‘¦π‘₯=π‘₯=π‘₯2𝑦=𝑦+𝑦2π‘₯=π‘₯+π‘₯𝑦=𝑦+𝑦2π‘₯=π‘₯+π‘₯2.andandand

3rd case:

Points 𝐴 and 𝐡 have neither the same π‘₯-coordinate nor the same 𝑦-coordinate, therefore π‘₯β‰ π‘₯ and π‘¦β‰ π‘¦οŒ οŒ‘.

Consider the point 𝐾 with coordinates (π‘₯,𝑦). Using the midpoint theorem, point 𝑀 midpoint of line segment 𝐴𝐡 and point 𝐺, midpoint of line segment 𝐴𝐾, lie on a straight line parallel to line ⃖⃗𝐡𝐾, therefore, parallel to the 𝑦-axis. Similarly, point 𝑀 midpoint of line segment 𝐴𝐡 and point 𝐸, midpoint of line segment 𝐡𝐾, lie on a straight line parallel to line ⃖⃗𝐴𝐾, therefore, parallel to the π‘₯-axis. We conclude that these two points have the same 𝑦-coordinate.

Points 𝐴 and 𝐾 having the same 𝑦-coordinate, then, the midpoint 𝐺 of line segment 𝐴𝐾 has π‘₯-coordinate of π‘₯+π‘₯2

Since points 𝑀 and 𝐺 have the same π‘₯-coordinate, we conclude that: π‘₯=π‘₯+π‘₯2

Points 𝐡 and 𝐾 having the same π‘₯-coordinate, then, the midpoint 𝐸 of line segment 𝐡𝐾 has 𝑦-coordinate of 𝑦+𝑦2

Since points 𝑀 and 𝐸 have the same 𝑦-coordinate, we conclude that: 𝑦=𝑦+𝑦2

The midpoint of line segment 𝐴𝐡 is therefore the point 𝑀π‘₯+π‘₯2;𝑦+𝑦2.

Note

The π‘₯-coordinate of the midpoint of a line segment is the arithmetic mean of the π‘₯-coordinates of the endpoints of this line segment and the 𝑦-coordinate of the midpoint of a line segment is the arithmetic mean of the 𝑦-coordinates of the endpoints of this line segment.

III. Distance between two points

Property

Consider an orthonormal coordinate plane (𝑂;𝐼, 𝐽).

Let a point 𝐴 with coordinates (π‘₯,𝑦); and a point 𝐡 with coordinats (π‘₯,𝑦).

Then the distance 𝐴𝐡 is given by the formula: 𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)

Proof

We consider in an orthonormal coordinate plane (𝑂;𝐼, 𝐽) two points 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) such that: π‘₯β‰ π‘₯ and π‘¦β‰ π‘¦οŒ οŒ‘.

We consider the point 𝐾 with coordinates (π‘₯,𝑦).

The triangle 𝐴𝐾𝐡 is a right-angled triangle at 𝐾. In fact, the straight line ⃖⃗𝐴𝐾 is parallel to the π‘₯-axis and the straight line ⃖⃗𝐡𝐾 is parallel to the 𝑦-axis.

We have 𝐴𝐾=(π‘₯βˆ’π‘₯) and 𝐡𝐾=(π‘¦βˆ’π‘¦)

Using the Pythagorean theorem in the triangle 𝐴𝐡𝐾 right-angled at 𝐾, we have: 𝐴𝐾+𝐡𝐾=𝐴𝐡𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)so

Finally, since a distance is a positive number, we have: 𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

Note

The above property is only valid in an orthonormal coordinate plane.

Example

Let two points 𝐴(5,βˆ’3) and 𝐡(2,7) in an orthonormal coordinate plane (𝑂;𝐼, 𝐽) where the unit length equals 1 cm.

The distance of the line segment 𝐴𝐡 is given by: 𝐴𝐡=√(2βˆ’5)+(7+3)=√109.

A value approximated to 10 of the length 𝐴𝐡 can be obtained using a calculator and we find π΄π΅β‰ˆ10.4cm.

Example 2: Midpoints and parallelograms

Statement

In an orthonormal coordinate plane (𝑂;𝐼, 𝐽), we consider the points 𝐷(βˆ’2,1), 𝐸(3,3), 𝐹(1,βˆ’1) and 𝐺(βˆ’4,βˆ’3).

Show that the quadrilateral 𝐷𝐸𝐹𝐺 is a parallelogram.

Hint: show that the diagonals of the quadrilateral 𝐷𝐸𝐹𝐺 have the same midpoint.

Answer

We calculate the coordinates of the midpoint of the line segments 𝐷𝐹 and 𝐸𝐺.

The midpoint of 𝐷𝐹 has the coordinates ο€Ό1βˆ’22,βˆ’1+12=ο€Όβˆ’12,0.

The midpoint of 𝐸𝐺 has the coordinates ο€Όβˆ’4+32,βˆ’3+32=ο€Όβˆ’12,0.

The diagonals of the quadrilateral 𝐷𝐸𝐹𝐺 have the same midpoint, so 𝐷𝐸𝐹𝐺 is a parallelogram.

Example 3: Distance and alignment

Statement

In an orthonormal coordinate plane (𝑂;𝐼, 𝐽), consider the points 𝐴(17,6), 𝐡(βˆ’10,βˆ’3) and 𝐢(8,3).

Are points 𝐴, 𝐡 and 𝐢 collinear?

Method: we calculate the lengths 𝐴𝐡, 𝐴𝐢, and 𝐢𝐡 then we use the triangular inequality.

Answer

Let us calculate the lengths 𝐴𝐡, 𝐴𝐢, and 𝐢𝐡. 𝐡𝐴=√(βˆ’10βˆ’17)+(βˆ’3βˆ’6)=9√10.𝐢𝐴=√(8βˆ’17)+(3βˆ’6)=3√10.𝐡𝐢=√(8+10)+(3+3)=6√10.

Triangular inequality

Therefore, we have 𝐡𝐴=𝐡𝐢+𝐢𝐴.

Points 𝐡, 𝐢 and 𝐴 are therefore collinear in this order.

Example 4: Solved exercise 1

Write an algorithm that requires from the user to enter the length of the side 𝑐 of a square and returns the perimeter 𝑃 and the area 𝐴 of that square.

Answer

The relation of the perimeter 𝑃 and the area 𝐴 as a function of the side 𝑐 of the square is: 𝑃=4×𝑐 and 𝐴=π‘οŠ¨

Example 5: Solved exercise 2

Write an algorithm that asks to find the coordinates of two points 𝐴 and 𝐡 and that returns the coordinates of the point 𝐼 midpoint of the segment 𝐴𝐡.

Answer

Let (π‘₯,𝑦) the coordinates of 𝐴 and (π‘₯,𝑦) the coordinates of 𝐡 and let the point 𝐼 the midpoint of the line segment 𝐴𝐡, then the coordinates of 𝐼 are written as: ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

Example 6: Solved exercise 3

In an orthonormal coordinate plane, we have the points 𝐴(βˆ’4,βˆ’4), 𝐡(7,βˆ’1), and 𝐢(βˆ’1,7).

What is the type of the triangle 𝐴𝐡𝐢?

Answer

If we plot the points in a coordinate plane, the triangle appears to be equilateral.

We calculate then the lengths of the three sides of the triangle 𝐴𝐡𝐢. 𝐴𝐡=√(7+4)+(βˆ’1+4)=√(11)+(3)=√130𝐴𝐢=√(βˆ’1+4)+(7+4)=√(3)+(11)=√130𝐡𝐢=√(βˆ’1βˆ’7)+(7+1)=√(βˆ’8)+(8)=√128

We get 𝐴𝐡=π΄πΆβ‰ˆ11.4, and π΅πΆβ‰ˆ11.31 so the triangle 𝐴𝐡𝐢 is isosceles at 𝐴.

Despite the impression that it gives, it is not equilateral because the length 𝐡𝐢 is different from the other two calculated lengths.

Example 7: Solved exercise 4

Given an orthonormal coordinate plane (𝑂;𝐼, 𝐽). Let 𝓁 the circle with center 𝐼 and radius 3.

Let 𝑀(π‘₯,𝑦) any point on the plane. Which relation between its coordinates π‘₯ and 𝑦 characterizes the fact that this point 𝑀 belongs to the circle 𝓁? Deduce if a point 𝐾 with coordinates (3,βˆ’2) satisfies this relation.

Answer

Saying that 𝑀 lies on the circle 𝓁 is to say that 𝐼𝑀=3 or 𝐼𝑀=9.

The point has coordinates (1,0) and the coordinate plane is orthonormal, we have 𝐼𝑀=9, hence, (π‘₯βˆ’1)+(π‘¦βˆ’0)=9.

A point 𝑀(π‘₯,𝑦) belongs to the circle 𝓁 if and only if (π‘₯βˆ’1)+𝑦=9.

By substituting the coordinates of point 𝐾 in the previous equation, we have: (3βˆ’1)+(βˆ’2)=8β‰ 9

So, point 𝐾 does not belong to the circle 𝓁.

Practical Work 1 (TP): Algorithm

Given the following algorithm:

  • What does this algorithm do?
  • Try this algorithm for points 𝐴(βˆ’1,0) and 𝐡(2,1) then give the corresponding values of 𝑛 and 𝑑.

Practical Work 2 (TP): Extraction from an excel sheet

Djibouti question

Each box, called β€œcell”, is identified by a letter and a number.

For example, cell F8 contains the number 50. It is then possible to write in a cell the content of another cell, to do that, we only have to write the identifier of this other cell and preceding it by a β€œ=”. Therefore, cell E8 contains the same information as cell D5.

After validation, we can read β€œ28” in cell E8.

  • One of the cells contains the formula =5+3βˆ’2ABA. Which cell is that?
  • Which cell contains the formula =6βˆ’8βˆ’5βˆ’2βˆ’8EBDAF?

Practical Work 3 (TP): Study of a configuration

Let 𝐴𝐡𝐢𝐷 a square and point 𝐼 the midpoint of the line segment 𝐴𝐡. We draw inside the square the equilateral triangle 𝐴𝐡𝐸 and outside the square the equilateral triangle 𝐡𝐢𝐹.

We want to prove that the points 𝐷, 𝐸, and 𝐹 are collinear.

  1. Draw the figure
  2. We choose as coordinate plane (𝐴;𝐡, 𝐷).
    1. Indicate the coordinates of points 𝐴, 𝐡, 𝐢 and 𝐷.
    2. Calculate the length 𝐼𝐸. Deduce the coordinates of point 𝐸.
    3. Determine the coordinates of point 𝐹.
  3. Calculate the lengths 𝐷𝐸, 𝐷𝐹, and 𝐸𝐹. What can we conclude from this?

Practical Work 4 (TP): Place points using a coordinate plane

Let two lines Ξ” and Ξ”β€² intersecting at 𝐴.

We place a point 𝐡 outside the lines Ξ” and Ξ”β€².

The objective of this practical work is to place a point 𝐸 on Ξ” and a point 𝐹 on Ξ”β€² such that point 𝐡 is the midpoint of the line segment 𝐸𝐹.

  1. Using trial and error:
    Redraw the figure on a sheet of paper or on a geometry software and try to determine the positions of points 𝐸 and 𝐹. Describe the process used.
  2. Geometric method:
    Redraw the figure and create a parallelogram 𝐴𝐹𝐻𝐸 according to the conditions of the statement. Describe the creation process of the parallelogram 𝐴𝐹𝐻𝐸. Deduce the positions of points 𝐸 and 𝐹 on the lines Ξ” and Ξ”β€².
  3. Using a coordinate plane:
    Using the coordinate plane (𝐴;𝐢, 𝐷), find the coordinates of points 𝐸 and 𝐹.
    Deduce their position on the lines Ξ” and Ξ”β€².

Key Points

  • Reading the coordinates of a point in a coordinate plane

    The point 𝑀 has the coordinates (2,4) in this oblique coordinate plane (𝑂;𝐼, 𝐽).
  • Calculating the coordinates of the midpoint of a line segment
    The midpoint of the line segment 𝐴𝐡 has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.
  • Triangular inequality
    Let 𝐴, 𝐡, and 𝐢 be three points of the plane, if 𝐡𝐢=𝐡𝐴+𝐴𝐢 then the points 𝐡, 𝐴, and 𝐢 are collinear in this order.
  • Calculating a distance between two points in an orthonormal coordinate plane
    If we know the coordinates of two points of the plane 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) in an orthonormal coordinate plane (𝑂;𝐼, 𝐽), the distance 𝐴𝐡 is given by the formula: 𝐴𝐡=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦).

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