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Math 7 – Chapter 8 Student Notes
Name: _____________________________________
Teacher: ___________________________________
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Activating Prior Knowledge
Classifying Triangles
Quick Review
Here are two ways to classify triangles.
 By side length
An equilateral triangle
has all sides equal.
An isosceles triangle
has 2 equal sides.
A scalene triangle
has all sides different.
A right triangle
has one 90° angle.
An obtuse triangle
has one angle greater
than 90°.
 By angle measure
An acute triangle
has all angles less
than 90°.
Check
Use square dot paper or triangular dot paper.
1. Draw an isosceles triangle.
Is it acute, obtuse, or right? How do you know?
2. Draw an obtuse triangle.
Is it equilateral, scalene, or isosceles? How do you know?
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3. Can you draw an obtuse isosceles triangle?
If you can, draw it.
If you cannot draw the triangle, say why it cannot be drawn.
4. Can you draw a right equilateral triangle?
If you can, draw it.
If you cannot draw the triangle, say why it cannot be drawn.
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Properties of a Rhombus
Quick Review
A rhombus is a parallelogram with all sides equal.
The properties of a rhombus are:
 Opposite angles are equal.

The diagonals intersect at right angles.
 Opposite sides are parallel.
 The diagonals bisect each other.
 Diagonals bisect the angles.
Example
Look at rhombus PQRS.
a) Name all pairs of parallel sides.
b) What is the measure of PCQ? How do you know?
c) What do you know about the measures of QRC
and SRC?
d) What do you know about the lengths of SC and CQ?
Solution
a) Opposite sides are parallel.
So, RQ is parallel to SP and SR is parallel to PQ.
b) The diagonals of a rhombus intersect at right angles.
So, PCQ = 90°
c) The diagonals of a rhombus bisect the angles.
So, QRC = SRC
d) RP is the perpendicular bisector of SQ.
So, SC = CQ
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Check
5. Use a ruler and a protractor.
Draw any rhombus ABCD.
Draw the diagonals of the rhombus.
Name:
i) all equal sides
ii) pairs of parallel sides
iii) all equal angles
iv) all right angles
v) the perpendicular bisector of AC vi) the perpendicular bisector of BD
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Parallel Lines
Parallel lines are lines that never meet. They are located on
the same flat 2-D surface and they are always an equal distance
apart.
Example:
There are many ways to draw parallel lines. One method involves
using a ruler and a compass. Using only a ruler and a compass,
draw a set of parallel lines.
STEPS
1. Draw a line segment (using a ruler). Label it AB. Mark
another point, C, not on AB.
2. Mark any point D on AB.
3. Place the compass on point D. Make sure the circle drawn
goes through point C. Label point E where the circle
intersects AB.
4. Do not alter the compass. Place the compass on point E and
draw a circle through point D.
5. Place the compass on point E and set the pencil point on C.
6. Without changing this distance, place the compass on point D
and draw a circle. Where this circle intersects the other, mark
point F.
7. Draw a line through C and F. CF will be parallel to AB.
AB // CF
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Example 2
Draw a line segment KL anywhere on your paper. Use a ruler
to ensure that KL is 7 cm long. Construct another line segment XY
so that XY // KL. You are only allowed to use a ruler and a
compass.
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Perpendicular Lines
Recall that parallel lines are lines that are always an equal
distance apart and they never meet or cross. Perpendicular lines
are lines that meet exactly once and they intersect at right angles
(they form a 90° angle).
Example:
As with parallel lines, there are many ways to construct
perpendicular lines. You can use rulers, compasses, protractors, or
even simple paper folding. We are going to use a ruler and a
compass to construct two perpendicular line segments.
STEPS
1. Using a ruler, draw line segment AB. Mark point C on AB.
2. Set the compass on point B and draw a circle that has a larger
radius than half of the distance between B and C. (Make sure
the circle intersects AB).
3. Without changing the compass’ distance, draw a second
circle using point C as the center. This circle should intersect
the first one in two locations. Mark these spots as point D and
E.
4. Draw a line (using your ruler) through points D and E. DE is
perpendicular to AB.
*there are other methods of drawing perpendicular lines using a compass.
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Example 2
Using a ruler, draw an 8 cm line segment, GH, anywhere on
your paper. Create a perpendicular line segment, JK, to GH using
only a ruler and a compass.
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Constructing Perpendicular Bisectors
When a line segment is divided into two equal pieces, you
have bisected it. The line that divides a line segment into these
equal pieces is called a bisector. If this bisector forms a right angle
with the line segment, it is called a perpendicular bisector.
bisector
perpendicular bisector
To construct a perpendicular bisector, you can use a MIRA, a
ruler, or even paper folding. We are going to use a compass.
STEPS:
1. Draw a line segment, AB.
2. Starting at point A, draw a circle that has a radius that is at
least one-half the distance of AB.
3. Without changing the compass, draw the same circle from
point B.
4. The two circles should intersect at two points. Label these
points C and D.
5. Draw a line through C and D (using a ruler). CD is
perpendicular to AB. Since ABCD is a rhombus (a fourequal sided 2-D shape) the diagonals (AB and CD) bisect
each other so CD is the perpendicular bisector of AB.
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Example 2
Using a ruler and a compass, draw the perpendicular bisector
of a line segment, MN, that is 9cm long.
Would you have to alter your process if the line segment was 1 cm
long? Explain.
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Constructing Angle Bisectors
An angle is formed when two lines, line segments or rays
intersect. Our goal is to learn how to divide an angle into two equal
parts. The line that cuts an angle into two equal parts is called an
angle bisector.
STEPS:
1. Draw an angle (of any size) and label it ABC . Make sure
that B is the angle’s vertex.
2. From point B draw a circle that cuts the angle in two places.
Label each intersection point D and E.
3. Without changing the compass starting at point D, draw a
circle (it should pass through point B)
4. Without changing the compass again, draw the same circle
from point E.
5. These two circles should intersect. Label the intersection
point F.
6. Connect B and F. BF is the angle bisector for angle ABC.
(ABCF creates a rhombus where AC and BF are diagonals,
so they bisect each other).
*no matter which construction you do, you always can check your
work by measuring angles with a protractor and line segments with
a ruler.
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Example 2
Using a protractor, draw an angle that measures 47° and an
angle that measures 135°. Construct the angle bisector for each of
the angles using only a compass and a ruler.
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Graphing on a Coordinate Plane
A coordinate plane (Cartesian Plane) is a grid system used
for plotting points. You have used a portion of the plane when you
have drawn bar or broken-line graphs. The coordinate plane looks
like:
y-axis
Quadrant II
Quadrant I
x-axis
origin
Quadrant III
Quadrant IV
Points are always written in the form (x, y). To plot these
points, you always plot the x-coordinate first. This means that you
always move left or right from the origin first then up or down.
Example: On the grid, plot A(3,1), B(-2,4) and C(0,-4).
What is the horizontal distance
between A and B?
What is the vertical distance
between B and C?
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Example 2
Draw out a coordinate plane that has a maximum of +5 (for
both x and y) and a minimum of –5 (for both x and y).
a.) Plot the points A(4, 2), B(-3, -1), C(4, -1) and D(-3, 2).
b.) Connect A to C, C to B , B to D and D to A. What shape is
it?
c.) Find the perimeter and area. Show your work!
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Graphing Translations and Reflections
An object that is graphed on a coordinate plane can be moved
right, left, up or down (called a translation), flipped (called a
reflection) or turned (called a rotation). Any movement or change
to an object is generally referred to as a transformation.
To complete a translation, we need to know the horizontal
and vertical change.
Example: On a grid, move points A(2,3), B(4,1) and (0,-1) four
units left and three units down.
*Notice that the original shape and its image are congruent; they are exactly
the same.
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To complete a reflection, we need to know the line in which
to reflect the image. This line is called to reflection (or mirror)
line.
Example: Flip ABC over the x-axis if A(0,3), B(2,4) and C(1,1).
Notice that the original shape and the image are congruent but have different
orientations (they face different directions).
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Example 2
Graph
KLM
where K(-1,-4), L(2,-1) and M(1,3).
a.) Translate the triangle 2 units right and 3 units down. Label it
ABC .
b.) Reflect the original image over the x-axis. Label it ABC .
c.) Reflect the original image over the y-axis. Label it
ABC .
Be sure to list the coordinates for all of the images.
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Graphing Rotations
A rotation is a turn around a point of rotation (or a turn
centre). The point of rotation can be on the object or off. The
rotation can be clockwise or counterclockwise.
Example: Draw quadrilateral ABCD, where A(5,3), B(1, 4), C(4,1)
and D(1,0). Rotate this object 90° clockwise about the point D.
Notice that the original object and the image are congruent and like
reflections, they have different orientations.
Example: Using the shape ABCD (from above), rotate it 180°
about the origin.
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Chapter Review
I should be able to:
 Define parallel lines, perpendicular lines, right angle,
bisector, perpendicular bisector, angle, angle bisector,
coordinate plane, x-axis, y-axis, origin, reflection, rotation,
translation, congruent, and orientation
 Use a compass and a ruler to construct parallel lines,
perpendicular lines, perpendicular bisectors and angle
bisectors
 Explain in words (with or without a diagram) how to
construct parallel lines, perpendicular lines, perpendicular
bisectors and angle bisectors
 Plot points in all four quadrants of a coordinate plane
 Label the x-axis, y-axis, origin and quadrants on a diagram
of the coordinate plane
 Translate, reflect, and rotate any 2-D object on a
coordinate plane
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Chapter 8: Practice Unit Exam
Name: _____________
MULTIPLE CHOICE
Select the best response for each of the following questions. [1 mark
each]
1. The point (-2, 3) would be located in which quadrant?
a. I
b. II
c. III
d. IV
2. Which statement is FALSE?
a. Parallel lines are always the same distance apart.
b. A perpendicular bisector will always cut a line segment in half.
c. A rotation is the only transformation that changes the
orientation of a shape.
d. All of these statements are true.
3. A point, P, has the coordinates (2, -8). What are its new coordinates
after reflecting point P over the y-axis?
a. (-2, 8)
b. (-2, -8)
c. (2, -8)
d. (2, 8)
4. After rotating point (1,3) 90° clockwise about the origin, the new
location is _________.
a. (-3,1)
b. (-1,3)
c. (1,3)
d. (3,-1)
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5. A translation of any object changes the _____________.
a. Size
b. Orientation
c. Location
d. Congruency
WRITTEN RESPONSE
Read each question carefully. Show all of your work.
6. In the space provided below, draw a line segment AB that is 5 cm
long. Construct another line segment, CD, so that AB // CD. Beside
your illustration, explain the steps for HOW to draw the construction.
[6 marks: 2 for the drawing, 4 for the explanation]
7. Add the following items to the coordinate plane below [1 mark each]
a. x-axis
b. y-axis
c. origin
d. point H(-3,1)
e. point G(2, -3)
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8. On the grid below, draw A(2, 5), B(4,0) and C(0,0). [1 mark]
a.) Translate ABC 2 units left and 3 units down. Be sure to label your
triangle. [3 marks]
b.) Reflect ABC over the x-axis. Be sure to label your triangle. [3 marks]
c.) Rotate ABC 270° clockwise about the origin. Be sure to label your
triangle. [3 marks]
d.) Translate ABC 1 unit right and 1 unit up and then reflect it over the yaxis. Be sure to label your triangle. [3 marks]
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9. The picture below shows three images of an arrow. Identify each
transformation. [1 mark each]
Original
1. _____________
1
2. _____________
3
3. _____________
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10.Find the horizontal distance between each of the following pairs of
points. [1 mark each]
a. A(-5,3) and B(4, 4)
b. C(3, -2) and D(0,5)
11.Find the vertical distance between each of the following pairs of
points. [1 mark each]
a. E(0,4) and F(6, -2)
b. G(8,9) and H(-2,-5)
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Important Terms
Term
Definition
Illustration
Triangle
Equilateral triangle
Isosceles triangle
Scalene triangle
Acute angle
Right angle
Obtuse angle
Rhombus
Congruent
Parallel lines
Perpendicular lines
Line segment
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Bisect
Bisector
Perpendicular bisector
Angle bisector
Coordinate grid
X – axis
Y – axis
Origin
Quadrants
Compass
Mira
Translation
Reflection
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Rotation
Prime
Ordered pair
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