Mathematics LE

Key Unit competence: To be able to determine algebraic representations of lines and calculate the area of

                                                   geometric shapes in 2D.

3.0 INTRODUCTORY ACTIVITY 3

RUKUNDO uses a number line to graph the points -4, -2, 3, and 4. Her classmate ISIMBI, notices that -4 is closer to zero than -2 as shown in the figure below. She told him that he is mistaken. Rukundo replied that his is not sure about his diagram because he did not have time to repeat his course yesterday evening.

a) Use what you know about a vertical number line to determine if RUKUNDO made a mistake or not. 

Support your explanation with a number line diagram

b) What is number line

c) What is Cartesian plane?

3.1. Cartesian coordinates of a point

ACTIVITY 3.1

Consider the points A(1, 2), B(5, 4)

1. Represents these points in xy plane

2. Draw a line segment from A to B

Content summary

A point is represented by Cartesian coordinates (also called rectangular coordinates). 

In two dimensions, Cartesian coordinates are a pair of numbers that specify signed distances from the coordinate axes. 

They are specified in terms of the x coordinates and the y coordinates. The origin is the intersection of the two axes.

The position of a point on the Cartesian plane is represented by a pair of numbers.

The pair is called an ordered pair or coordinates (x, y). The first number, x,

called the x-coordinate and the second number, y, is called the y-coordinate.

The origin is indicated by the ordered pair or coordinates (0, 0).

To get to the point (x, y) in cartesian plane, we start from the origin. If x is positive then we move x units right from the origin otherwise if x is negative then we move x units left from the origin. Then, if y is positive, we move y units up otherwise if y is negative, we move y units down

Example:

Represent the points M (2,1.5) , L(−3,1.5) and N (−2,−3) in Cartesian plane.

Solution

• Point M has coordinates M (2,1.5). To get to point M, we move 2 units to the right of x-axis from the origin (positive side) and 1.5 units up on y-axis from the origin (positive side).
•  Point L is represented by the coordinates L(−3,1.5) . To get to point L, we move 3 units to the left of x-axis from the origin (negative side ) and 1.5 units up on y-axis from the origin (positive side)
•  Point N has coordinates N (−2,−3). To get to point N, we move 2 units to the left of x-axis from the origin of(negative) and 3 units down on y-axis from the origin (negative side).

APPLICATION ACTIVITY 3.1

Represent the points A(−3,2) , B(4, 2) , C(−3,−2) , D(3,−1) in xyplane

3.2. Distances between two points

ACTIVITY 3.2

Kalisa and Mugisha live in the same Sector but at

two different hills. If Kalisa’s house is located at A(1,2) from the Sector and Mugisha’s house is at A(4,6) .

a) If the office of the sector is considered as the origin, present this situation on Cartesian plan.

b) Use the ruler to calculate the distance between Kalisa’s house and Mugisha’s.

Content summary

Recall from the Pythagorean Theorem that, in a right triangle, the hypotenuse c and sides a and b are related by a² + b² = c² . Conversely, if a² + b² = c² the triangle is a right (see figure below).

If the two points do not lie on a horizontal or on a vertical line, they can be used

to form a right triangle, as shown in in the figure below.

Examples:

1) Consider the points A(1,4), B(−2,−3) in Cartesian plane. Find the distance between the point A and B.

Solution

The distance is

2) Consider the points C(k,−2) and D(0,1) in cartesian plane. Find the distance between the point A and B.

APPLICATION ACTIVITY 3.2

1. Calculate the distance between the points given below:

         a) S(−2;−5)        and Q(7;−2)

         b) A(2;7)             and B(−3;5)

         c) A(x;y)               and B(x+4;y−1)

2. The length of CD = 5. Find the missing coordinate if:

            a) C(6;−2)          and D(x;2).

            b) C(4;y)            and D(1;−1).

3.3. Midpoint of a line segment

ACTIVITY 3.3

Three friends: Pascal, Steve, and Benjamin live on the same side of a street from Nyabugogo to Ruyenzi.

 Steve’s house is halfway between Pascal’s and Benjamin’s houses. If the locations of Pascal can be given by the coordinates (2,5) , Benjamin’s house at (4,7);

a) Draw a Cartesian plan and locate Pascal’s and Benjamin’s houses

b) Show the line segment joining the locations of Pascal’s and Benjamin’s houses;

c) Given that Steve’s house is in the half way, locate his house, estimate  the coordinate of that location

     and explain how to find it.

Content summary

The midpoint of a segment is a point on that line segment which maintains the same distance from both 

 of the endpoints of that line segment.

For example, consider segment to the left.

It has endpoints named A and B. The midpoint of the segment is labeled M. 

It is at the same distance from each of the endpoints.

The coordinates of the midpoint, are obtained using the midpoint formula, such that:

Example:

1) Find the midpoint of the segment joining points A(3,0) and B(1,8) .

Solution

APPLICATION ACTIVITY 3.3

Micheal and Sarah live in different cities and one day they decided to meet up for lunch. 

Because they both wanted to travel as little as possible they decided to meet at a point halfway between their homes. 

If their positions are given by (3100,500) and (5120,125).

Which of the following coordinates represents the place where they

should meet?

a) (4110,312.5)

b) (4110,375)

c) (2020,375)

d) (8220,625)

3.4. Vector in 2D and dot product

3.4.1. Vectors in 2D

ACTIVITY 3.4.1

In xy plane:

1. Represent the points A(1,2) and B(−3,1)

2. Draw arrow from point A to point B

Content summary

Definitions and operations on vectors

A vector is a directed line segment; it is a quantity that has both magnitude and direction.

 A vector is represented by an arrow.

The length of the arrow represents the magnitude of the vector, and the arrowhead indicates the direction of the vector.

 That is to say, a vector has a given length and a given direction.

Components of a vector

We now present an alternative representation of a vector in the plane that is common in the physical sciences. Let i ^→ 

denote the unit vector whose direction is along the positive x-axis;

APPLICATION ACTIVITY 3.4.1

1) In xy plane, present the following vectors:

2) Plot a vector with initial point P(1,1) and terminal point Q(8,5) in the Cartesian plane and illustrate its position vector.

3.4.2 Dot product

ACTIVITY 3.4.2

Content summary

Scalar product and properties

The scalar product or dot product (or sometimes inner product) is an algebraic operation that takes two coordinate vectors and returns a single number.

Algebraically, it is the sum of the products of the corresponding coordinates of the two vectors.

We can illustrate this scalar product in terms of work done by a force on the body:

Suppose that a person is holding a heavy weight at rest. This person may say and feel he is doing hard work but in fact none is being done on the weight in the scientific sense. Work is done when a force moves its point of application along the direction of its line of action. A force F can act on a body and move it to a displacement^→S .

Properties of scalar product

APPLICATION ACTIVITY 3.4.2

1. Find the norm(magnitude) of the following vectors:

3.5. Equation of a straight line

3.5.1. Equation of a straight line passing through a point and parallel to a direction vector

ACTIVITY 3.5.1

APPLICATION ACTIVITY 3.5.1.

1. Find the vector, parametric and Cartesian equation of the line passing through the point with position vector (2,−3) 

and parallel to the line (x, y) = (3,5) + r (1,6)

2. Find the Cartesian equations of the lines whose vector equations are given below. 

Give your answers in the form y = mx + c

3.5.2 Equation of a straight line given 2 points

ACTIVITY 3.5.2

Content summary

Therefore,

APPLICATION ACTIVITY 3.5.2

1) Determine the equation of a straight line passing through P(2, 4) and B(3,−2) .

3.5.3 Equation of a straight line given its gradient

LEARNING ACTIVITY 3.5.3

Determine the slope and the -intercept in the equations below. Hence draw the lines in the same graph.

Consider a line having gradient and passing through the point P₁ ( X₁,Y₁) .

Suppose that the point P(x, y) is an other point on the line. 

Then the gradient of the line is the rate at which the line rises (or falls) vertically for every unit across to the right. 

It is defined by the change in to the change in .

Thus, the equation of the line in point-slope form is defined by

From the equation above, if we take any other point P₂ (X₂,Y₂ )  that lies on this line, then:

Parametric equations

Cartesian equation

Example :

Write the equation of the line that has slope that passes through the

Solution

Note:

The genaral equation of the line is Ax + By +C = 0, where A, B,C are constantes,

And, A ≠ 0, B ≠ 0 .

• If A = 0 the line is horizontal.
•  If B = 0 the line is vertical.
•  If C = 0 the line passes through the origin.

• • If the two lines are parallel, then their slopes/gradients are equal.

     Therefore  m₁ = m₂

Thus, the equations Ax + By +C = 0 and Ax + By + D = 0 are parallel.

•  If two lines are peripendicular, then the product of their slopes/ gradients is equal to −1.
•  Therefore  m ₁×m₂ = −1.
•  Thus, the equations Ax + By +C = 0 and Bx − Ay + D = 0 are peripendicular.

Example :

1) Find the equation for the line that contains the point (5,1) and is parallel to

2) Find the equation for the line that contains the point (0, −2) and is perpendicular to y = 5x + 3?

Step1: Identify the slope of the given line and write its negative reciprocal.

APPLICATION ACTIVITY 3.5.3

1. Write the equation in point-slope form for the line through the given point that has the given slope

2. Is y − 5 = 2(x −1) an equation of a line passing through(4, 11) ? Explain.

3. Write an equation of the line that contains the point (−3,−5) and the same slope as y + 2 = 7(x + 3)

3.6. Problems on points and straight lines in 2D

3.6.1 Intersection, perpendicularity or parallelism of two lines

ACTIVITY 3.6.1

1. In the same Cartesian plane plot the straight lines containing the following points

What is your opinion about the two lines in the Cartesian plane?

2. In the same Cartesian plane plot the straight lines containing the following points (−3,−1); (0,3) and ( − 2,−4); (1,0)

a) Perpendicularity and intersection of two lines

Here L₁ and L₂ are perpendicular

To write a straight line perpendicular to a given straight line we proceed as follows:

Step I: Interchange the coefficients of x and y in equation ax + by + c = 0.

Step II: Interchange the sign between the terms in x and y of equation i.e., If the coefficient of x and y in the given equation are of the same signs make them of opposite signs and if the coefficient of x and y in the given equation are of the opposite signs make them of the same sign.

Step III: Replace the given constant of equation ax + by + c = 0by an arbitrary constant.

For example, the equation of a line perpendicular to the line 7x + 2y + 5 = 0 is

2x − 7y + c = 0 again, the equation of a line, perpendicular to the line9x − 3y =1 is 3x + 9y + k = 0

Note:

Assigning different values to k in bx − ay + k = 0 , we shall get different parallel straight lines each of which is 

perpendicular to the line ax + by + c = 0. Thus we can have a family of straight lines perpendicular to a given straight line.

Notes:

To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of x and y so obtained determine the coordinates of the point of intersection.

Solution:

Example

Find the equation of the straight line which is parallel to 5x − 7y = 0 and passing through the point (2,−3) .

Solution:

The equation of any straight line parallel to the line 5x − 7y = 0 is 5x − 7y +λ = 0

………… (i) [Where λ is an arbitrary constant].If the line (i) passes through the

point (2,−3) then we shall have:

Therefore, the equation of the required straight line is 5x − 7y + 31 = 0

d) Angles between two lines

We can illustrate the scalar product in terms of work done by the force on the body:

If the force does not act in the direction in which motion occurs but an angle θ to it , then the work done is defined as the product of the component of the force in the direction of motion and the displacement in that direction.

Notice

• Two lines are perpendicular if the angle between them is a multiple of a right angle
• Two lines are parallel and with the same direction if the angle between them is a multiple of a zero angle
• Two lines are parallel and with the opposite direction if the angle between them is a multiple of a straight angle

Example:

Solution

Let α be the angle between these two vectors

APPLICATION ACTIVITY 3.6.1

1. Find the equation of a straight line that passes through the point

(−2,3) and perpendicular to the straight line 2x + 4y + 7 = 0

2. Find the equation of the straight line which passes through the point

of intersection of the straight lines x + y + 9 = 0 and 3x − 2y + 2 = 0 and is perpendicular to the line

4x + 5y +1 = 0

3. Find the equation of the straight line passing through the point

(5,−6) and parallel to the straight line 3x − 2y +10 = 0 .

3.6.2. Distance between two lines and the distance between a point and line

ACTIVITY 3.6.2

1. In one Cartesian plane plot the straight lines containing the following points

Distance between a point and a line

The perpendicular distance from a point D(x₁, y₁) to the line ax + by + c = 0 is given by

Example

Find the shortest distance from the line 2x + y −1 = 0 to the point (3,2)

Solution :

Distance between two parallel lines and two intersecting lines

Example

What is the distance between the lines3x + 4y = 9 and 6x + 8y =15 ?

Solution:

First of all we find the slopes of the two lines. We convert the equations into slope intercept form

APPLICATION ACTIVITY 3.6.2

3.7 Geometric shapes in 2D

3.7.1 Identification of the Geometric shapes in two dimensions

LEARNING ACTIVITY 3.7.1

1) i) Discuss the difference between regular polygons and irregular

polygons and give examples.

ii) Name the shapes with three sides and give at least 4 examples of

materials in real life with three sides.

iii) Name the shapes with four sides and give at least 4 examples of

materials in real life with four sides.

2) i) Plot the following points on the 2D coordinate plane, then use a

ruler to connect the points in the order they are listed to form a polygon.

1) i) Discuss the difference between regular polygons and irregular polygons and give examples.

ii) Name the shapes with three sides and give at least 4 examples of materials in real life with three sides.

iii) Name the shapes with four sides and give at least 4 examples of materials in real life with four sides.

2) i) Plot the following points on the 2D coordinate plane, then use a

ruler to connect the points in the order they are listed to form a polygon.

Polygon 1: (8,6);(11,1);(8,6)

Polygon 2: (2,5);(6,5);(6,3);(2,3)

Polygon3: (3,5);(7, 2);(7,−5);(3,−2)

Polygon4: (−2,5);(−2,−2),(3,−2),(5,5)

Polygon5: ( 2, 2);(2, 2);(2, 2);(6, 2),(− − − 6,−6);(−4,−6);(−4,−4);(−2, 4)

ii) What is the name of each geometric figure formed?

iii) Describe the similarities and differences of those figures.

Content summary

In geometry, a two-dimensional shape can be defined as a flat plane figure or

a shape that has two dimensions – length and width. Length on x- axis and width on y-axis or vice versa

Two-dimensional or 2-D shapes do not have any thickness and can be measured

in only two faces. In the two –dimensional coordinate, geometric figures can be plotted:

Example : The following points;

a) (8,6);(11,1);(8,6)            b) (2,5);(6,5);(6,3);(2,3)

Can make the following polygons in coordinate’s graphs

Two dimensional shapes are for example: Triangle, square, rectangle,

parallelogram, trapezium, rhombus, circle, pentagon, hexagon, etc.

It is also important to take note of the hierarchy, e.g. all rectangles are

parallelograms but all parallelograms are not rectangles.

Example:

Explain why square is rectangle but a rectangle is not a square?

Solution:

Depending on their properties, a square is a quadrilateral with four right angles

and equal sides. The two parallel sides are obviously equal, which allows a

square to be a rectangle. Even though the two parallel sides of a rectangle are

equal, the four sides are not equal; which means that a rectangle is not a square

but a square is rectangle.

APPLICATION ACTIVITY 3.7.1

3.7.2 Perimeter and area of geometric shapes in 2D.

LEARNING ACTIVITY 3.7.2

1) The following fields have two dimensional shapes, find the length of

one wire that can be used on one row of the fence in order to protect the plants.

2) On the graph below, one square of the paper represents one square

unit. Calculate the area of given shapes in the graph

Content summary

As studied in previous levels (Primary and ordinary levels), the perimeter and

area of two dimensional shapes can be summarised in the following table:

Example

Find the perimeter and the area of the regular octagon below. A regular octagon

has all sides that are equal.

Irregular polygons

If the shape is irregular, it can be divided into regular or other simple figures

to find its area. while the perimeter is found by adding the sum of sides for the original figure.

Examples: Given that one square of the paper represents one square unit, find the area of shaded figures.

APPLICATION ACTIVITY 3.7.2

Two different copies of a geometric figure were provided as follow:

i) Given that one square of the paper represents one square unit on each figure,

find the area of the geometric figure ABCDE in blue.

ii) Show and explain that different methods can be used when finding the area of an irregular polygon.

3.8 END UNIT ASSESSMENT

1) A new park is being designed for your city. The plans for the design

are being drawn on the coordinate plane below. The vertices given

form a polygon that represents the location for 5 different features in the park.

a) Plot each set of points in the order they are given to form a polygon.

Label the feature that the polygon represents on the graph.

Playground:

Splash Pad: (10, 4),(−4,8),(6,8)

Restrooms: (−4,5),(−4,8),(−7,10),(−10,7),(−7,5)

Picnic Pavilion: (8,−1),(6,−3),(4,−3),(2,−1),(2,1),(4,3),(6,3),(8,1) .

If each square on the graph represents 1 square meter, answer the questions that follow.

b) Cement needs to be poured to lay the foundation for the restroom and

the splash pad. How many square meters of cement will the city need

for the foundation of these two features?

c) The playground is going to be covered with wood chips, how many square

meters of wood chips will the city need for the playground?

d) The citizens of the city have asked that the playground have a fence

around its perimeter. How many meters of fencing will they need for the playground?

e) The foundation of the pavilion picnic area is going to be covered with

special pavers. How many square meters of pavers will they need for the pavilion?

2) A coordinate grid represents the map of a city. Each square on the grid represents one city block.

a) Helena’s apartment is at the point . She walks 4 blocks south, then 8 blocks west, then 4 blocks north, 

    and then finally 8 blocks east back to her apartment. How many blocks did she walk total?

Describe the shape of her path. Mark and label her apartment and highlight her walk.

b) Draw and describe in words at least two different ways you could walk exactly 20 blocks and 

      end up back where you came from.

c) Carl lives at the point , Find the distance between Helena’s house and Carl’s house, and then mark and label Carl’s house.

d) Establish the equation of the line passing through the points H(5,7) and C(5,-5). 

Given that one block has 10 m of length’s side, calculate the distance between them, 

and the area of the square centered in C with radius .

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