chapter 7 coordinate geometry 7.1 midpoint of the line joining two points 7.2 areas of triangles and...
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Chapter 7
Coordinate Geometry
7.1 Midpoint of the Line Joining Two Points
7.2 Areas of Triangles and Quadrilaterals
7.3 Parallel and Non-Parallel Lines
7.4 Perpendicular Lines
Coordinate Geometry
Objectives
7.1 Midpoint of the Line Joining Two Points
In this lesson, you will learn how to find the midpoint of a line segment and apply it to solve problems.
A line AB joins points (x1, y1) and (x2, y2).
M (x, y) is the midpoint of AB.
y
x
A ( x1 , y1)
B ( x2 , y2)y
x
A ( x1 , y1)
B ( x2 , y2)
M ( x , y)
y
x
A ( x1 , y1)
B ( x2 , y2)
M ( x , y)
C ( x2 , y1)
y
x
A ( x1 , y1)
B ( x2 , y2)
M ( x , y)
C ( x2 , y1)D ( x , y1)
E ( x2 , y)
y
x
A ( x1 , y1)
B ( x2 , y2)
M ( x , y)
C ( x2 , y1)D ( x , y1)
E ( x2 , y)
Construct a right angled triangle ABC.
Construct the midpoints D and E of the line segments AC and BC. Take the mean of the coordinates at the endpoints.
D is and E is1 21,
2
x xy
1 21, 2
y yx
M is the point 1 2 1 2,2 2
x x y y
Take the x-coordinate of D and the y-coordinate of
E.
Coordinate Geometry
y
x
S ( a , b)
Q ( 9 , 6)
R ( – 2 , 4)
P ( 4 , – 4)
O
4 2 4 4, 1,0
2 2M
P, Q, R and S are coordinates of a parallelogram and M is the midpoint of PR. Find the coordinates of M and S and show that PQRS is a rhombus.
y
x
S ( a , b)
Q ( 9 , 6)
R ( – 2 , 4)
P ( 4 , – 4)
O
M
y
x
S ( a , b)
Q ( 9 , 6)
R ( – 2 , 4)
P ( 4 , – 4)
OM ( 1 , 0)
y
x
S ( a , b)
Q ( 9 , 6)
R ( – 2 , 4)
P ( 4 , – 4)
OM ( 1 , 0) 9 6
, 1,02 2
a bM
7, 6a b = 7, 6S
M is also the midpoint of QS.
229 4 6 4 125PQ
2 29 2 6 4 125QR
the parallelogram is a rhombusPQ QR PQRS
y
x
S ( – 7 , – 6)
Q ( 9 , 6)
R ( – 2 , 4)
P ( 4 , – 4)
OM ( 1 , 0)
Coordinate Geometry
Example 4
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
1 3 6 2, 1,4
2 2D
3 points have coordinates A(–1, 6), B(3, 2) and C(–5, –4). Given that D and E are the midpoints of AB and AC respectively, calculate the midpoint and length of DE.
1 5 6 4, 3,1
2 2E
Let M be the midpoint of DE.
2 23 1 1 4 5DE
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
D
E
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
D ( 1 , 4)
E
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
D ( 1 , 4)
E ( – 3 , 1)
12
1 3 4 1, 1,2
2 2M
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
D ( 1 , 4)
E ( – 3 , 1) M
y
x
B ( 3 , 2)
A ( – 1 , 6)
C ( – 5 , – 4)
D ( 1 , 4)
E ( – 3 , 1)
M ( – 1 , 2 12 )
Coordinate GeometryExercise 7.1, qn 3
y
x
B ( p , – 2)
D ( 3 , r )
A ( 2 , 0)
C ( – 1 , 1)
1 12 2
2 1 0 1, ,
2 2M
1 12 2
23, ,
2 2
rpM
2, 3p r
Let M be the midpoint of AC.
If A(2, 0), B(p, –2), C(–1, 1) and D(3, r) are the vertices of a parallelogram ABCD, calculate the values of p and r.
y
x
B ( p , – 2)
D ( 3 , r )
A ( 2 , 0)
C ( – 1 , 1)M
y
x
B ( p , – 2)
D ( 3 , r )
A ( 2 , 0)
C ( – 1 , 1)M ( 1
2 , 1
2 )
M is also the midpoint of BD.
y
x
B ( p , – 2)
D ( 3 , r )
A ( 2 , 0)
C ( – 1 , 1)M ( 1
2 , 1
2 )
y
x
B ( – 2 , – 2)
D ( 3 , 3 )
A ( 2 , 0)
C ( – 1 , 1)M ( 1
2 , 1
2 )
Coordinate Geometry
Exercise 7.1, qn 4