Exercise Distance

1. Refer to Certesian plane as shown find the distance between:
(a) U and V

(b) U and W




2. Given that the coordinationes of P,Q,R and S are (1,1),(4,5),(-1,-1) and (-3,2) respectively, find the distance between
(a) P and Q

(b) R and Q

3. Refer to the above situation. What is the final distance between the Empire spaceship and Enterprise spaceship?



Solution

Exercise Identify coordination

1.      Plot the following points on a cartesian plane.

a.       A(5,2)   B(-3,2)  C(4,-1)  D(-4,-4)

b.      P(4,2)   Q(5,-1)  R(-4,-2)  S(-3,5)

2.      The diagram shows a cartesian plane.

a.       State the coordination of points R,S,T and U.

                      Solution             

Exercise Midpoint

1.     Refer to the Cartesian Plane as shown and find the coordinates of thr midpoint of the line

a)     PQ

b)    QR

1.     Find the coordinates of the midpoint of the straight line joining point P (-2, 2) and point Q (4 ,6).

Solution

The coordinates of the midpoint of the line PQ are ( 1, 4)

1.     Refer to the Cartesian plane as shown and find the coordinates of the midpoint of the straight line joining

a)     P and F                                                                              

b)    H and J

c)     D and  I

d)    E and F

e)     G and H

f)      I and J

1.     Find the coordinates of the midpoint of the line which joins the following pairs of points.

a)     P (2, 5) and Q (8, 5)

b)    M (5, -2) and M (-1, -2)

c)     J (3, -4) and K (3, 2)

d)    A (-4 ,1) and B (-4,-3)

2.     Refer to the Cartesian plane as shown and find the coordinates of the midpoint of the straight line joining

a)     R and Q

b)    T and S

c)     T and U

d)    S and Q

1.     Find the coordinates of the midpoint of the line which joins the following pairs of  points

a)     P (6, 1) and Q ( 2,7)

b)    M (3, 4) and N (-1, 6)

c)     J (2, 5) and K (6, 1)

d)    A (-3, 2) and B (-1, -8)

2.     Given that ( 2, p) is the midpoint of the straight  line joining the points ( 1, 8) and (3, -2), find the value of p

3.     Given that (4 ,3) is the midpoint of the straight line joining the points (k, -1) and (2, 7), find value of k

4.     PQRS is the rhombus. Given that the coordinates of P, Q and R are (-3, 2), (1, 4) and (5, 2) respectively.

a)     Find the coordinates of S

b)    Find the coordinates of midpoint of all the sides of the rhombus.

c)     What the shape of the straight lines which  join all the midpoint?

SOLUTION

           1.

a)    The coordinates of the midpoint of the line PQ are (3, 4).

b)    The coordinates of the midpoint of the line QR are ( 5, 1).

2.  The coordinates of the midpoint of the line PQ are ( 1, 4)

2.   a) ( 0, 4)    b)(-1, 1)

c) (-2, -3)   d)(2, 1)

e) (-3, 0)    f)(1, -1)

      4.  a)(5,5)    b)(2, -2)

           c) (3, -1)  d)(-4, -1)

      5. a) (-1, 3)  b) ( 3, 1)

           c) ( -2, 0) d) (3, 4)

      6. a) (4, 4)  b) ( 1, 5)

        c) ( 4, -2 )  d) ( -2 , -3 )

      7. P = 3

     8. K = 6

     9.  a) ( 1, 0)

          b) (-1, 3), (3 , 3),(3, 1),(-1,1)

          c) Rectangle

Midpoint

Learning Outcomes :

 1. Identify the midpoint of a straight line joining two points.

2. Find the coordinates of the midpoint of a straight line joining two points with :

                               I.            Common y-coordinates.

                            II.            Common x-coordinates.

3. Find the coordinates of the midpoint of the line joining two points.

4. Pose and solve problem involving midpoints.

Activity

Aim           : to find the midpoint of a straight line

Instruction : Carry out this activity in groups of four.

Materials   : Tracing paper, ruler and pencil.

Procedure :

1.      Draw a straight line on a piece of tracing paper, Label the line as PQ.

2.      Fold the tracing paper so that the two ends of the line overlap each other perfectly.

3.      Unfold the tracing paper, mark the folded part of the line as R.

4.      Is R equidistant from P and Q ? Discuss.

From the activity , we find that R is midpoint of the line PQ.

Example

 Identify the midpoints of the following straight lines.

a)

SOLUTION

Points C is midpoint of AE.

b)                 

SOLUTION

Points O is the midpoint of MQ.

Distances between two point

Distance between two point in A Cartesian Plane.

1. Find the distance between two points with :

                               I.            Common y coordinates.

                            II.            Common x coordinates.

There are three ways to find the distance between two point:

a)      By inspection

For example, in the Cartesian Plane as shown, the distance between a and b is 4 units. The distance between c and d is 3 units.

b)      By moving one point to another

For example, in a Cartesian Plane as shown A has to move for units to reach B. therefore, the distance  between A and B is 4 units. C has to move 3 units up to reach D. therefore, the distance between C and D is 3 units.

c)     Finding the difference between the x coordinate or y coordinate :

for example the distance between A and B

= difference between the x coordinate

= 5-1

= 4 units.

The distance between C and D

= difference between the y coordinates

= -1- (-4)

= -1 + 4

= 3 units.

l. find the distance between two points using Pythagoras’ theorem.

The Cartesian plane shows the positions of two aircrafts, A and B. we can find the distance between the two aircrafts by drawing and appropriate right – angled triangle using AB as hypotenuse.

Hence, AB²  = AC² + CB²  

                            = 3² + 4²

                        = 25.

                  AB = 5 units

Thus, the distance between the aircraft is 5 units.

video tutorial

Coordinate Plane

Midpoint

identify coordinates

Learning outcomes :

1.      Identify the x-axis, y-axis and the origin in the Cartesian Plane.

In the Cartesian Plane , the horizontal number line is called the x-axis whereas the vertical number line is called the y-axis. The intersection point of the  x-axis and the y-axis is known as the origin is usually represented with the letter 0

2.      Plotting point and stating the coordinates of the points.

The  position of any point Cartesian Plane can be determined by its distance from each of the axis .

For example if  point P in the Cartesian Plane is located by an ordered pair(a,b), then

                               I.            The x-coordinate of P is a and the distance of P from the y-axis is a units.

                            II.            The y-coordinate of P is b and the distance of P from the x-axis is b units.

3.      Plotting point and stating the distances of the points from the y-axis and x-axis.

Example.

The coordinate of P are (4,3)

4.      Stating the coordinates of points in a Cartesian plane.

The Cartesian plane has four quadrants.

In quadrant  1, the coordinates of any point are positive.

In quadrant  2, the x-coordinate of any point is negative

In quadrant 3, the coordinates of any point are negative.

In quadrant 4, the y-coordinates of any point are negative.

For example, point A is 2 units to the right of the y-axis and 3 units above the axis. Therefore, its coordinates are (2,3). Point B is 3 units to the y-axis and 3 unit above the axis. Therefore, its coordinates are (-3, 3). State the coordinates of point C and point D.

scale x-axis y-axis

Scales for the coordinate Axis.

Learning outcomes :

1.  Mark the values on both axis by extending the sequence of given values on the axis.

2.  State the scales used in given coordinate axis where:

                                 I.            Scale for the axis are the same.

                              II.            Scale for the axis are difference.
3.  Mark the values on both axis, with reference to the scales given.
4. state the coordinate of a given point, with reference to the scales given.
5. plot point , given the coordinate, with reference scales given.
6. pose and solve problem involving the coordinate.


Example :
State the coordinates of point P and point Q in each of the Cartesian plane shown below.

a)

        

b)

SOLUTION:

a) The coordinates of P are (15,8)            --> P is 15 units to the right of the y-axis, 8 units above                                                                                       the x-axis

     The coordinates of Q are (-10, -8)      --> Q is 10 units to the left of the y-axis, 8 units below                                                                                          the  x-axis

b) The coordinates of P are (-9, 40)          --> P is 9 units to the left of the y-axis, 40 units above the

                                                                       x-axis.

     The coordinates of Q are (6,40).          --> Q is 6 units to the right of the y-axis, 40 units below                                                                                     the x-axis.

Introduction Coordination

What are coordinates?

To introduce the idea, consider the grid on the right. The columns of the grid are lettered A,B,C etc. The rows are numbered 1,2,3 etc from the top. We can see that the X is in box D3; that is, column D, row 3.

D and 3 are called the coordinates of the box. It has two parts: the row and the column. There are many boxes in each row and many boxes in each column. But by having both we can find one single box, where the row and column intersect.

The Coordinate Plane

In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two scales - one running across the plane called the "x axis" and another a right angles to it called the y axis. (These can be thought of as similar to the column and row in the paragraph above.) The point where the axes cross is called the origin and is where both x and y are zero.

On the x-axis, values to the right are positive and those to the left are negative.
On the y-axis, values above the origin are positive and those below are negative.

A point's location on the plane is given by two numbers,the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15. These are the coordinates of the point A, sometimes referred to as its "rectangular coordinates". Note that the order is important; the x coordinate is always the first one of the pair.