UNIT 2 – ANALYTIC GEOMETRY
Lesson
§
2.1
2.1
2.2
2.2
2.3
2.3
2.4
2.12.3
Feb.
28
2.5
2.12.3
Mar. 1
2.6
2.4
Mar. 2
2.7
2.5
Mar. 3
2.8
2.6
Mar. 6
2.9
2.7
Mar.
7/8
2.10
Mar. 9
2.11
Date
Feb.
22
Feb.
23
Feb.
24
Feb.
27
TOPIC
Homework
Midpoint of a Line Segment
Pg. 78 # 1, 3, 4 - 6, 8, 18
Length of a Line Segment
In Class Assignment - Problems
Equation of a Circle
Pg. 86 # 1, 3, 5ii, 6, 7, 9, 17
Equations of Altitudes/Bisectors and
Medians
Equations of Altitudes/Bisectors and
Medians
Classifying Figures on a Cartesian Plane
Quiz (2.1 – 2.4)
Verifying Properties of Geometric Figures
Exploring Properties of Geometric Figures
- Centroid/Circumcentre/Orthocentre
Using Coordinates to Solve Problems
Quiz (2.5 – 2.7)
Review for Unit 2 Test
TEST- UNIT 2
Pg. 91 # (1 – 6)doso, 7i, iii, 8, 10, 13,
14, 18
WS 2.4 # 1ac, 2, 3bd, 5, 6aceg
Pg. 79 # 7, 13ac
WS 2.5 # 1ac, 2, 3bd; # 1ace, 4;
# 1ace, 2a
Pg. 101 # 1 – 3, 7, 8, 9a, 12, 17
Pg. 109 # 2, 3, 5, 8, 14
Pg. 120 # 9 – 12, 14
Pg. 120 # 16, 20, 21
Pg. 124 # 1 – 3, 5, 7 – 11, 13, 15, 17,
18, 19a, 21 – 23, 25
MPM 2D
GEOMETRY DEFINITIONS
TRIANGLES
Equilateral – a triangle with 3 sides of equal length and 3 angles of equal measure.
Isosceles – a triangle with 2 sides of equal length. The angle opposite the equal sides are also of
equal measure.
Scalene – a triangle with no equal sides or angles.
Right – a triangle with one 90° angle.
QUADRILATERALS
Parallelogram – a quadrilateral where opposite sides have are parallel.
Rectangle – a parallelogram in which all interior angles are 90°.
Rhombus – a parallelogram with all sides of equal length.
Square – A rectangle with 4 sides of equal length
F
MEDIAN – a line that joins the vertex of a triangle to
the midpoint of the opposite side.
FT is one median of ∆DEF
E
T
PERPENDICULAR BISECTOR – A line that is perpendicular to another line segment and passes through the
midpoint of the line segment. ie: In ∆ABC, the line segment from M is the
perpendicular bisector of CB.
A
C
M
B
D
ALTITUDE – the line segment representing the height of a polygon, drawn from a vertex perpendicular
to the opposite side.
P
– PS is the altitude of ∆PQR.
R
Q
S
MIDSEGMENT – the line segment formed by joining the midpoints of 2 adjacent sides of a polygon
Midsegment
CENTROID –
the centroid of a triangle can be found by finding the equations of 2 of the median lines,then
finding the point of intersection of those two lines. It can also be found by calculating the
average of the x- and y-coordinates of all three vertices of the triangle. Diagram Pg. 112.
CIRCUMCENTRE – can be found by finding the equations of the perpendicular bisectors of 2 sides of a
triangle, then finding the point of intersection of those two lines. Diagram Pg. 112.
ORTHOCENTRE – can be determined by finding the equations of 2 of the altitude lines, then finding
the point of intersection of those two lines. Diagram Pg. 112.
PERPENDICULAR LINES – have slopes that are the negative reciprocal of each other.
ml  m 1l
1
2
PARALLEL LINES – have equal slopes.
ml  ml
1
2
SLOPE – Y-INTERCEPT FORM OF EQUATION OF A LINE
 y  mx  b
m
y 2  y1
x 2  x1
STANDARD FORM OF EQUATION OF A LINE  Ax  by  C  0 , where A, B, C  R
MPM 2D
Lesson 2.1
Midpoint of a Line Segment
Find the midpoint of the line segment below.
y
6
5
4
3
2
1
–6
–5
–4
–3
–2
–1
–1
1
2
3
4
5
6
x
–2
–3
–4
–5
–6
The coordinates of the midpoint of a line segment are the averages of the endpoints of the line segment.
A ( x1 , y1 )
x  x 2 y1  y2 
,

M  1
 2
2 
B ( x2 , y2 )
 to find the midpoint given the two endpoints, x M 
where ( x M , y M ) is the midpoint.
Ex. 1
y  y2
x1  x2
and y M  1
,
2
2
Find the coordinates of the midpoint of the line segment with endpoints A(–9, 3) and B(5, 7).
Ex. 2
One end of a line segment is P(–6, 3) and the midpoint is M(4, –1). Find the other endpoint Q.
Ex. 3
A circle has a radius with endpoints E(2, 4) and F(-4, -8). Find two possible endpoints for the
diameter that contains the radius.
There are 2 possible circles.
B
E
E
F
I. Let B be the other endpoint of the diameter.
F
A
II. Let A be the other endpoint of the diameter.
Pg. 78 # 1, 3, 4 - 6, 8, 18
MPM 2D
Lesson 2.2
Length of a Line Segment
To find the distance between two points, we can use the Pythagorean Theorem.
y
x
Ex. 1 Find the length of the line segment connect the two points A and B below.
y
6
5
4
B
3
2
1
–6
–5
–4
–3
–2
–1
–1
1
2
3
4
5
6
x
–2
–3
–4
A
–5
–6
Ex. 2 Find the distance between P(-3, 7) and (7, -8).
Ex. 3 A plane is flying from Toronto to Halifax. When at coordinates (125, 309) the plane develops engine
trouble. Ottawa International Airport is at coordinates (97, 215) and Pierre Elliot Trudeau Airport
is at coordinates (139, 412). To which airport should the plane divert?
Pg. 86 # 1, 3, 5ii, 6, 7, 9, 17
MPM 2D
Lesson 2.3
EQUATION of a CIRCLE with CENTRE (h, k)
CIRCLE – the set of all points in a plane that are the same distance from a fixed point, the centre.
The distance from the centre of a circle to any point on the circle is called the radius.
If the centre of the circle is at the point (h, k) and the radius is r units, then the equation
(x – h)2 + (y – k)2 = r2 is the equation of the circle.
(h, k)
.
Ex. 1 Write the equation of the circle with centre (0, 0) and radius of
1
units.
2
Ex. 2 A circle with centre (0, 0) passes through the point (6, -8).
a) Find the equation of the circle.
b) What is the other endpoint of the diameter that passes thru (6, -8)?
Ex. 3 A stone is dropped in the water and sends out ripples whose radius increases at 5 cm/s. Find the
equation of the outer ripple 12 s after the stone is dropped.
Ex. 4 Find the equations of the following circles.
a) centre (-3, 2) with a radius of 4
b) centre ( 2, -5) passing through the point ( 9, 10)
Ex. 5 Given: (x – 1) + (y + 3) = 50
Determine if the following points are inside, on, or outside the circle.
2
a) (-2, 7)
2
b) (4, 1)
c) (6, 2)
Ex. 4 A truck with a wide load is approaching a tunnel that is shaped like a semicircle. The maximum height of
the tunnel is 5.25 m high. The load is 8 m wide and 3.5 m high. The driver uses his grade 10 math skills
to determine if the load will it fit through the tunnel. Must the driver take another route?
Pg. 91 # (1 – 6)doso, 7i, iii, 8, 10, 13, 14, 18
MPM 2D
Lesson 2.4
Equations of Altitudes, Bisectors and Medians
To find the equation of a line you need:
 the slope of the line
 two points on the line
OR
 a point on the line
Ex. 1 Find the equation of the line through A(2, 3) and B(3, 5).
General Form of Equation of a
Line:
y  mx  b
Standard Form of Equation of a
Line:
Ax  By  C  0, A, B, C  I , A  0
Ex. 2 Find the equation of the line through P(5, 4) and perpendicular to AB, for A(-1, 3) and B(6, -4).
Ex. 3 Find the equation of the line through Q(-1, 0) and parallel to 2 x  3 y  1  0 .
Ex. 4 ABCD has vertices A(3, 7), B(-4, -2), C(6, 1) and D(-1, 0). Find the equation of the midsegment of
sides AB and BC.
WS 2.4 # 1ac, 2, 3bd, 5,
6aceg
MPM 2G
Lesson 2.5
Equations of Altitudes, Bisectors and Medians
F
MEDIAN – a line that joins the vertex of a triangle to
the midpoint of the opposite side.
FT is one median of ∆DEF
E
D
T
Ex. 1 If ABC has vertices A(12, 4), B(-6, 2), and C(-4, -2), find the equation of the median from B.
PERPENDICULAR BISECTOR – A line that is perpendicular to another line segment and passes through the
midpoint of the line segment. ie: In ∆ABC, the line segment from M is the
perpendicular bisector of CB.
A
C
M
B
Ex. 2 ABC has vertices A(12, 4), B(-6, 2), and C(-4, -2), find the equation of the perpendicular bisector of AB.
ALTITUDE – the line segment representing the height of a polygon, drawn from a vertex perpendicular
to the opposite side.
P
– PS is the altitude of ∆PQR.
R
S
Q
Ex. 3 ABC has vertices A(-1, 4), B(-1, -2), and C(5, 1), find the equation of the altitude to A.
Pg. 79 # 7, 13ac
WS 2.5 # 1ac, 2, 3bd; # 1ace, 4; # 1ace, 2a
MPM 2D
Lesson 2.6
SHAPE
Classifying Figures on a Cartesian Plane
DEFINITION
SKETCH
What is required in order
to classify the shape
Scalene Triangle A 3-sided polygon
where there are
no sides of equal
length.
Find lengths of all sides.
Isosceles
Triangle
A 3-sided polygon
which has 2 sides
of equal length.
Find lengths of all sides.
Equilateral
Triangle
A 3-sided polygon
which has 3 sides
of equal length.
Find lengths of all sides.
Right Triangle
A 3-sided polygon
which has one 90°
angle.
-find the slopes of all
sides
-show that 2 slopes are
negative reciprocals of one
another
Parallelogram
A quadrilateral
with opposite
sides that are
parallel.
-find the slopes of all
sides
-show that opposite sides
have the same slope
Rectangle
A parallelogram in
which all the
angles are 90°.
Square
A rectangle in
which all sides
are of equal
length.
-find the length of all
sides
-show opposite sides have
the same length
-show that adjacent sides
have slopes are negative
reciprocals
-find lengths of all sides
-find slopes of sides
-only need to prove one
90° angle
Rhombus
A parallelogram in
which all sides
are of equal
length.
-find the length of all
sides
-if all sides are equal, it
is a rhombus - a square is
a special rhombus
Ex. 1 Show that quadrilateral EFGH with vertices E (–5, 4), F (–2, 8), G (6, 2), H (3, –2) is a rectangle.
Ex. 2 Given: A (–1, 3), B (1, 7), and C (5, 5) a) verify that ABC is a right triangle.
b) determine whether or not ABC is isosceles.
Ex. 3 Determine the type of quadrilateral defined by the following vertices.
A(2, 3), B(5, -1), C(10, -1), D(7, 3)
Pg. 101 # 1 – 3, 7, 8, 9a, 12, 17
MPM 2D
Lesson 2.7
Verifying Geometric Properties
Ex. 1 Show that the midsegments of quadrilateral PQRS with vertices P(-2, -2), Q(0, 4), R(6, 3)
and S(8, -1) form a parallelogram.
y
9
8
7
6
5
Q
4
R
3
2
1
–9
–8
–7
–6
–5
–4
–3
–2
–1
–1
–2
P
–3
–4
–5
–6
–7
–8
–9
1
2
3
4
5
6
7
8
S
9
x
Ex. 2 a) Show that A(-4, 3) and B(3, -4) lie on the circle x + y = 25.
2
2
b) Show that the perpendicular bisector of chord AB passes through the centre of the circle.
a chord is the line segment joining
two points on a curve. The term is
often used to describe a line
segment whose ends lie on a circle.
Pg. 109 # 2, 3, 5, 8, 14
MPM 2D
Lesson 2.8
Exploring Properties of Geometric Figures
- Centroids/Circumcentres/Orthocentres
CENTROID – the centroid of a triangle can be found by finding the equations of 2 of the median lines,then
finding the point of intersection of those two lines. It can also be found by calculating the
average of the x- and y-coordinates of all three vertices of the triangle. It is the centre of
mass of the triangle. Diagram Pg. 112.
CIRCUMCENTRE – can be found by finding the equations of the perpendicular bisectors of 2 sides of a
triangle, then finding the point of intersection of those two lines. It is the point which is
equidistant from each vertices. Diagram Pg. 112.
ORTHOCENTRE – can be determined by finding the equations of 2 of the altitude lines, then finding
the point of intersection of those two lines. Diagram Pg. 112.
Ex. 1 For the triangle with vertices A(4, 9), B(1, 1), and C(8, -4).
a) Find the centroid.
b) Find the circumcentre.
c) Find the orthocentre.
Pg. 120 # 9 – 12, 14
MPM 2D
Lesson 2.9
Using Coordinates to Solve Problems
Ex. 1 The closest power line to a parking lot runs along a line through the points A(0, 4) and B(12, 10). At what
point should contractors connect a power line so that they use the least amount of cable to reach a
lamppost at coordinates L(6, 19)? How much cable, correct to one decimal place, will they use?
Pg. 120 # 16, 20, 21