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Chapter 3
Geometry and Measurement
What You Will Learn:
To identify, describe, and draw: Parallel line segments Perpendicular line segments
To draw: Perpendicular bisectors Angle bisectors
Generalize rules for finding the area of: Parallelograms Triangles
Explain how the area of a rectangle can be used to find the area of: Parallelograms Triangles
3.1 – Parallel and Perpendicular Line Segments
What you will learn:To identify, describe, and draw:
Parallel line segmentsPerpendicular line segments
Parallel
Describes lines in the same plane that never cross, or intersect
The perpendicular distance btw parallel line segments must be the same at each end of the line segments.
They are always marked using “arrows”
http://www.mathopenref.com/parallel.html
Some ways to create parallel line segments:
Using paper foldingUsing a ruler and a right triangle
Example:Draw a line segment, AB. Draw another line
segment, CD, parallel to AB.
Example:Draw a line segment, AB. Draw another line segment,
CD, parallel to AB.B
AC
D
B
AC
D
Label the endpoints (A, B, C, D).Mark the lines with arrows to show the lines are parallel.
B
A
Use a ruler todraw a linesegment.
Slide the triangle, draw a parallel line.
Perpendicular
Describes lines that intersect at right angles (90°)
They are marked using a small square
http://www.mathopenref.com/perpendicular.html
right angle
Using paper folding (p. 85)Using a ruler and protractor (p. 85)http://www.mathopenref.com/
constperplinepoint.html
Some ways to create perpendicular line segments:
Assignment
P. 86#1, 3-5, 7, 9, 11, Math LinkStill Good? #2, 8, 10, 12, 13ProStar? #14-16
right angle
3.2 – Draw Perpendicular Bisectors
Bisect:Bi means “two.” Sect means “cut.” So, Bisect
means to cut in two.
Perpendicular bisectorA line that divides a line segment in half and is
at right angles (90°) to the line segment.Equal line segments are marked with “hash”
marks
Some ways to create a perpendicular bisector:
Using a compass (p. 90)http://www.mathopenref.com/constbisectline.html
Using a ruler and a right triangle (p. 91)Using paper folding (p. 91)
Assignment
P. 92, # 1-5, 8Still Good? # 6, 7, 9, MathLinkProStar? #10
3.3 – Draw Angle Bisectors
Terms:Acute angle
An angle that is less than 90°Obtuse angle
An angle that is more than 90°Angle Bisector
A line that divides an angle into two equal partsEqual angles are marked with the same symbol
Less than 90°
Greater than 90°
Some ways to create an angle bisector include:
Using a ruler and compass (p. 95)http://www.mathopenref.com/constbisectangle.html
Using a ruler and protractor (p. 95)Using paper folding (p.95)
Assignment
P. 97, # 1 & 2, 5, 6, 8Still Good? # 3 & 4, 9, 11, 13, MathLinkProStar? #12, 14, 15
Greater than 90°: obtuse
Less than 90°: acute
Angle Bisector
3.4 – Area of a Parallelogram
Area of a rectangle: Area = length x width
ParallelogramA four-sided figure with opposite sides parallel
and equal in length
http://www.mathopenref.com/parallelogramarea.html
w
l
6 cm
4 cmA = l x w
A = 6 cm x 4 cm
A = 24 cm2
Making a Parallelogram from a Rectangle
cut
paste
BaseA side of a two-dimensional closed figureCommon symbol is b
HeightThe perpendicular distance from the base to the
opposite sideCommon symbol is h
Suggest a formula for calculating the area of a parallelogram.
b
h
Area of a Rectangle vs. Area of a Parallelogram
Area = length x width = 12 cm x 8 cm = 96 cm
Area = base x height = 12 cm x 8 cm = 96 cm
12 cm
8 cm
12 cm
8 cm
Are they the same? Try it!
2 2
b
h
Sometimes it is necessary to extend the line of the base to measure the height
Key Ideas
The formula for the area of a rectangle can be used to determine the formula for the area of a parallelogram.
The formula for the area of a parallelogram is A = b x h, where b is the base and h is the height.
The height of a parallelogram is ALWAYS perpendicular to its base.
h
b
Assignment
P. 104, # 1-3, 5, 7, 9, 11Still Good? # 13-18, MathLinkProStar? # 19, 20
b
h A = b x h
3.5 – Area of a Triangle
What you will learn:Develop the formula for the area of a triangleCalculate the area of a triangle
What we know:The area of a rectangle
A = l x w
The area of a parallelogramA = b x h
Key Ideas
The formula for the area of a rectangle or parallelogram can be used to determine the formula for the area of a triangle
The formula for the area of a triangle is A = b x h 2, or A = b x h, 2
where b is the base of the triangle and h is the height of the triangle. The height of the triangle is always measured perpendicular to its
base. http://www.mathopenref.com/trianglearea.html
h
b
A = b x h
h
b
Cut the rectangle in half
A = b x h 2
Cut the area in half
Your Assignment
P. 113, #1-3 as a class.Area of a Triangle, NotebookArea of a Triangle Questions, NotebookP. 113, #4a), 5b)No problem? #8, 10, 11Still good? #13-15Pro Star? #16-19