Courseware
Wednesday 11 December 2013
Friday 6 December 2013
Exercise midpoint
1.
Refer
to the Cartesian Plane as shown and find the coordinates of thr midpoint of the
line
a)
PQ
b)
QR
2.
Find
the coordinates of the midpoint of the straight line joining point P (-2, 2)
and point Q (4 ,6).
3.
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
4.
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)
5.
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
6.
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)
7.
Given
that ( 2, p) is the midpoint of the
straight line joining the points ( 1, 8)
and (3, -2), find the value of p
8.
Given
that (4 ,3) is the midpoint of the straight line joining the points (k, -1) and (2, 7), find value of k
9.
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?
Mistakes
When student answer the question mathematic especially subtopic coordination,they always do mistakes when answer the question.some of the mistakes that students do is :
Example 1:
The diagram above show points E, F and H on a Cartesian plane. If the midpoint of FH and EG is the same point, find the coordinates of G.
mistake 1
Join point E to the midpoint of FH. Extrapolate the straight line until it is of equal length to the line joining point E and the midpoint. Point G is the point where the line ends.
Therefore, the coordinates of point G is (4, 3)
Example 1:
The diagram above show points E, F and H on a Cartesian plane. If the midpoint of FH and EG is the same point, find the coordinates of G.
mistake 1
Join point E to the midpoint of FH. Extrapolate the straight line until it is of equal length to the line joining point E and the midpoint. Point G is the point where the line ends.
Therefore, the coordinates of point G is (4, 3)
Exercise Scale
1. Plot
the points P(6,8), R(-3,-4) and Q(12,-4) in a Cartesian
plane wherethe scale of the x-axis is 1 : 3 and the scale of the y-axis is
1 : 2.
2. Plot
the points P(6,9), R(-4,-3) and Q(12,-6) in a Cartesian
plane wherethe scale of the x-axis is 1 : 2 and the scale of the y-axis is
1 : 3.
3. Plot
the points P(4,15), R(0,-10) and Q(12,-10) in a Cartesian planewhere
the scale of the x-axis is 1 : 4 and the scale of the y-axis is 1 : 5.
4. Plot
the points P(-5,25), R(-15,10) and Q(25,40) in a Cartesian planewhere
the scale of the x-axis is 1 : 5 and the scale of the y-axis is 1 : 10.
5. Plot
the points P(6,8), R(-4,-4) and Q(12,-4) in a Cartesian
plane wherethe scale of the x-axis is 1 : 2 and the scale of the y-axis is
1 : 4.
Thursday 5 December 2013
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
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.
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.
Identifying the Midpoints of Straight Lines
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.
Identify the midpoints of the following straight lines.
a)
Identify the midpoints of the following straight lines.
a)
Points C is midpoint of AE.
b)
Points O is the midpoint of MQ.
The Midpoint of a Straight Line Joining Two Points with a Common y-
coordinate
In the above diagram, the point (0, 2) is the midpoint of the straight line joining
points A(-2, 2) and B(2,2).
example:
Find the coordinates of the midpoint of a line joining points A(3, 6) and B(-7, 6).
answer:
The Midpoint of a Straight Line Joining Two Points with a Common x-
coordinate
In the diagram on the right, point (1, 0) is the midpoint of the straight line joining
points A(1, 2) and B(1, -2).
example:
Find the coordinates of the midpoint of the line joining points P(1, 0) and Q(1, - 4).
answer:
x-coordinate for the midpoint = 1
Coordinates of the Midpoint of a Line Joining Two Points
example1:
Find the coordinates of the midpoint of the line joining point M(-1, 5) to point N(3,- 2).
answer:
example 2:
Find the coordinates of the midpoint of the line joining F(8, 2) and G(0, -6).
answer:
Distances Between Two Point
Distance between two point in A Cartesian Plane.
1. Find the distance between two points with :
1. Find the distance between two points with :
I. Common y coordinates.
II. Common x coordinates.
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)
= 3 units.
d) find the distance between two points using Pythagoras’ theorem.
d) 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.
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.
Scale for the coordinate axes refer to the ratio which shows the values on the axes that is represented by one units.
Scales Used on the Coordinate Axes
1 unit on the x-axis represents 2 units.
1 unit on the y-axis represents 2 units.
Therefore, the scale for x and y-axis is 1 : 2.
1 unit on the x-axis represents 5 units.
1 unit on the y-axis represents 3 units.
Therefore, the scale for the x-axis is 1 : 5 and the scale for the y-axis is 1 : 3.
Example :
State the coordinates of point P and point Q in each of the Cartesian plane shown below.
a)
solution:
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 the x-axis.
b)
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.
Values on the x-axis and y-axis Based on Given Scales for Both Axes
Example:
Mark the values on the x-axis and the y-axis on a Cartesian plane if the scale for the x-axis is 1 : 3 and
the scale for the y-axis is 1 : 5.
*the scale for the x-axis is 1:3
this is show 1 units represents for 3 units
*the scale for the y-axis is 1:5
this is show 1 units represents for 5 units.
solution
Stating the Coordinates of a Point Based on Given Scales for Both Axes
Example:
State the coordinates of each point marked on the Cartesian plane.
P(-5, 20)
Q(0, 14)
R(8, - 10)
S(-7, -12)
Plotting Points with Given Coordinates Based on Given Scales
Example:
Plot points A(1, 5) and B(-2, 20) on a Cartesian plane if the scale for the x-axis is 1 : 1 and the
scale for the y-axis is 1 : 5.
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