Exercise 1
Consider the number line below and the point midpoint of line segment .
- What is the -coordinate of each of the points , , , , and ?
- Read the -coordinate of the point symmetric to point with respect to the point .
Exercise 2
Consider the orthonormal coordinate plane , .
- Which of the following points , , , , and have:
- The same -coordinate?
- The same -coordinate?
- Give the coordinates of the points , , , , and .
- Is the following statement true or false?
“In the above figure, there are as many points that have a positive -coordinate as there are points that have a negative -coordinate.”
Exercise 3
Consider a triangle right-angled at such that:
and .
1. What is the exact length of line segment ?
Consider a triangle such that:
; and .
2. Is this triangle right-angled?
Activity 1: Grid
- Plot on a grid the points , , and as shown below.
- Plot the points , , and such that the point has coordinates in the coordinate plane , .
- What are the coordinates of points and in this coordinate plane?
Activity 2: Coordinates of midpoint
- In an orthonormal coordinate plane , , plot the points , , and .
- Read the coordinates of points and the midpoints of line segments and respectively.
- What is the relation that exists between the coordinates of point and the coordinates of points and ?
- Does this relation remain valid for point ?
Activity 3: Length of a line segment
- In an orthonormal coordinate plane , with 1 cm of unit length, place the points and .
- To find the distance in this coordinate plane:
- Plot the point . What is the type of triangle ?
- Find the distances and . Deduce the distance .
- Plot the point .
Calculate the distance 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 , .
Point is the origin of the coordinate plane, the line is the -axis and the line is the -axis.
The length of line segment is the unit length of the -axis and the length of the line segment is the unit length of the -axis.
Different types of coordinate planes
- If the axes and are intersecting at and are non-perpendicular, then , is called an oblique coordinate plane.
- If the axes and are perpendicular at and if , then , is called an orthogonal coordinate plane.
- If the axes and are perpendicular at and if, in addition, , then , 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 “checked” by an opponent’s piece if he is found on a square that can be reached by this piece.
- Locate the squares containing the pieces of this chess game.
- A player wants to move the white knight located at d8. Identify the possible squares that can contain this knight.
- The rook moves vertically or horizontally as many squares as needed. Where to put the rook to “check” the black king?
Answer
- 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.
- The white knight located at d8 can reach one of the following squares: c6, e6, b7, and f7.
- In order to “check” 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 , of non collinear points.
For a point in the plane:
- is the -coordinate on the axis of the point of intersection of axis with the line parallel to axis going through .
- is the -coordinate on the axis
of the point of intersection of axis
with the line parallel to axis going through .
In the coordinate plane , , a point is plotted using the ordered pair . The real numbers and are called the coordinates of point in the coordinate plane , .
Notation
We write to say that the real number is the -coordinate of point and the real number is the -coordinate of point .
Example
- The coordinate plane , in the figure is orthonormal.
- The point has coordinates .
- The origin has coordinates .
- The point has coordinates .
- The point has coordinates .
II. Coordinates of midpoint of a line segment
Property
Let , a coordinate plane, and two points with coordinates and respectively.
The midpoint of line segment has coordinates .
Example
Given the points with coordinates, and .
The midpoint of line segment has coordinates that is .
Proof
Consider the two points and in an orthonormal coordinate plane , .
is the midpoint of line segment .
1st case:
Points , and have the same -coordinate.
We have and .
So,
2nd case:
Points , and have the same -coordinate.
We have and .
So,
3rd case:
Points and have neither the same -coordinate nor the same -coordinate, therefore and .
Consider the point with coordinates . Using the midpoint theorem, point midpoint of line segment and point , midpoint of line segment , lie on a straight line parallel to line , therefore, parallel to the -axis. Similarly, point midpoint of line segment and point , midpoint of line segment , lie on a straight line parallel to line , therefore, parallel to the -axis. We conclude that these two points have the same -coordinate.
Points and having the same -coordinate, then, the midpoint of line segment has -coordinate of
Since points and have the same -coordinate, we conclude that:
Points and having the same -coordinate, then, the midpoint of line segment has -coordinate of
Since points and have the same -coordinate, we conclude that:
The midpoint of line segment is therefore the point .
Note
The -coordinate of the midpoint of a line segment is the arithmetic mean of the -coordinates of the endpoints of this line segment and the -coordinate of the midpoint of a line segment is the arithmetic mean of the -coordinates of the endpoints of this line segment.
III. Distance between two points
Property
Consider an orthonormal coordinate plane , .
Let a point with coordinates ; and a point with coordinats .
Then the distance is given by the formula:
Proof
We consider in an orthonormal coordinate plane , two points and such that: and .
We consider the point with coordinates .
The triangle is a right-angled triangle at . In fact, the straight line is parallel to the -axis and the straight line is parallel to the -axis.
We have and
Using the Pythagorean theorem in the triangle right-angled at , we have:
Finally, since a distance is a positive number, we have: .
Note
The above property is only valid in an orthonormal coordinate plane.
Example
Let two points and in an orthonormal coordinate plane , where the unit length equals 1 cm.
The distance of the line segment is given by: .
A value approximated to of the length can be obtained using a calculator and we find .
Example 2: Midpoints and parallelograms
Statement
In an orthonormal coordinate plane , , we consider the points , , and .
Show that the quadrilateral is a parallelogram.
Hint: show that the diagonals of the quadrilateral have the same midpoint.
Answer
We calculate the coordinates of the midpoint of the line segments and .
The midpoint of has the coordinates .
The midpoint of has the coordinates .
The diagonals of the quadrilateral have the same midpoint, so is a parallelogram.
Example 3: Distance and alignment
Statement
In an orthonormal coordinate plane , , consider the points , and .
Are points , and collinear?
Method: we calculate the lengths , , and then we use the triangular inequality.
Answer
Let us calculate the lengths , , and .
Triangular inequality
Therefore, we have .
Points , and 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 of a square and returns the perimeter and the area of that square.
Answer
The relation of the perimeter and the area as a function of the side of the square is: and
Example 5: Solved exercise 2
Write an algorithm that asks to find the coordinates of two points and and that returns the coordinates of the point midpoint of the segment .
Answer
Let the coordinates of and the coordinates of and let the point the midpoint of the line segment , then the coordinates of are written as: .
Example 6: Solved exercise 3
In an orthonormal coordinate plane, we have the points , , and .
What is the type of the triangle ?
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 .
We get , and so the triangle is isosceles at .
Despite the impression that it gives, it is not equilateral because the length is different from the other two calculated lengths.
Example 7: Solved exercise 4
Given an orthonormal coordinate plane , . Let 𝓁 the circle with center and radius 3.
Let any point on the plane. Which relation between its coordinates and characterizes the fact that this point belongs to the circle 𝓁? Deduce if a point with coordinates satisfies this relation.
Answer
Saying that lies on the circle 𝓁 is to say that or .
The point has coordinates and the coordinate plane is orthonormal, we have , hence, .
A point belongs to the circle 𝓁 if and only if .
By substituting the coordinates of point in the previous equation, we have:
So, point 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 and then give the corresponding values of and .
Practical Work 2 (TP): Extraction from an excel sheet
Each box, called “cell”, 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 “=”. Therefore, cell E8 contains the same information as cell D5.
After validation, we can read “28” in cell E8.
- One of the cells contains the formula . Which cell is that?
- Which cell contains the formula ?
Practical Work 3 (TP): Study of a configuration
Let a square and point the midpoint of the line segment . We draw inside the square the equilateral triangle and outside the square the equilateral triangle .
We want to prove that the points , , and are collinear.
- Draw the figure
- We choose as coordinate plane , .
- Indicate the coordinates of points , , and .
- Calculate the length . Deduce the coordinates of point .
- Determine the coordinates of point .
- Calculate the lengths , , and . What can we conclude from this?
Practical Work 4 (TP): Place points using a coordinate plane
Let two lines and intersecting at .
We place a point outside the lines and .
The objective of this practical work is to place a point on and a point on such that point is the midpoint of the line segment .
- 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 and . Describe the process used. - Geometric method:
Redraw the figure and create a parallelogram according to the conditions of the statement. Describe the creation process of the parallelogram . Deduce the positions of points and on the lines and . - Using a coordinate plane:
Using the coordinate plane , , find the coordinates of points and .
Deduce their position on the lines and .
Key Points
- Reading the coordinates of a point in a coordinate plane
The point has the coordinates in this oblique coordinate plane , . - Calculating the coordinates of the midpoint of a line segment
The midpoint of the line segment has the coordinates . - Triangular inequality
Let , , and be three points of the plane, if then the points , , and 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 and in an orthonormal coordinate plane , , the distance is given by the formula: .
