What was the main idea?
Look back at Explorations
9.4, 9.5, 9.6, 9.7, 10.7, 10.11, 10.12.
Pick 4 of these. Write down 1 or 2 main
concepts or conclusions.
For example: EX. 9.4: The line of reflection
is the perpendicular bisector of any
segment connecting corresponding image
and preimage points.
Main ideas 9.4
• The line of reflection is the perpendicular
bisector of any segment connecting
corresponding image and preimage points.
• Corresponding image and preimage points
are the same distance from the line of
reflection.
• Reflections produce congruent images.
Main ideas 9.5
• When folded as directed, the image
showing on the folded “quarter” is the
preimage of the lower right quadrant.
• The folds are the horizontal and vertical
lines of reflection.
• The unfolded images have at least 2
lines of symmetry.
Main ideas 9.6
• 180˚ rotation symmetry (point symmetry) is
different from reflection symmetry.
• Corresponding points on the image and
preimage are the same distance from the
point of rotation.
• The resulting image is congruent to the
preimage.
Main ideas 9.7
• Regular figures have rotational symmetry
and reflection symmetry. For an n-sided
regular figure, rotational symmetry is found
by 360/n, and there are n lines of reflection.
• Isosceles triangles, kites, isosceles
trapezoids, rhombi, rectangle and square
have reflection symmetry. Parallelograms
have only point (180˚rotational) symmetry.
Main ideas 10.7
• A triangle has area that is half of the
rectangle with the same base and height.
• Sometimes you need to find a bigger area,
and subtract out the parts you don’t need.
• Area is measured in square units.
• Area of a parallelogram is found by
translating the right triangle from one side to
the other, forming a rectangle.
Main ideas 10.11
• We can find the area of an irregular figure by
– cutting and pasting to make a new shape that is
close to the other,
– cutting one shape and filling in the other,
– making a grid,
– using non-standard units.
• Knowing the perimeter does not necessarily
help us find the area.
Main ideas 10.12
• If area is constant, then “long” or “skinny”
rectangles will have larger perimeter than
“square-like” rectangles.
• If perimeter is constant, then “long” or
“skinny” rectangles will have less area than
“square-like” rectangles.
• Knowing the perimeter does not always help
with finding the area of a figure.
• Perimeter and area are different! Perimeter
is distance around, and measured in units;
area is square units inside.
Agenda
• Have you learned the main ideas of chapters
9 and 10?
• Review computing perimeter and area of 2Dimensional regions.
• Review nets and solids
– Prisms and cylinders
• Assign homework
Pythagorean Theorem
• The most proved theorem ever--over
300 proofs! One was done by James
Garfield, before he was president of the
United States.
• If you have a right triangle with
hypotenuse of length “c”, then
a2 + b2 = c2.
It looks like this!
• a2 + b2 = c2.
But we use it like this.
• Find the perimeter and area of this
triangle.
5 feet
13 feet
x feet
Other ways to make our
life easy.
• Compare the circumference and area.
r
2r
Try this--find perimeter
and area
13 “
13 “
10 “
10 “
20 “
• P = tri + rect + sem
13 + 13 + 10 + 20 + 10
13 “
+ sem (.5 • 2π• 5)
• A = tri + rect + sem
52 + x2 = 132
x = 12
10 “
.5•10•12 + 20•10
+ .5•π•52
13 “
10 “
20 “
Try to find the shaded area
• Assume the
trapezoid
is
isosceles.
38 cm--whole base
7 cm
4 cm
24 cm
24 cm
•
•
•
•
Area of trapezoid - area of parallelogram
38 cm--whole base
Trap: .5 • 24 (24 + 38)
7 cm
Para: 7 • 4
4 cm
Did not need
Pythagorean
Theorem!
24 cm
24 cm
Find the perimeter and
area…
• If it looks right or congruent, it is.
• (1)
(2)
4 m2
14 m
18 in.
9 in. 9 in.
2.8 m
10 m
18 in.
2m
One
• Perimeter
– Sides of large
triangle: 92 + 92 = x2
x = 12.7
9 in.
12.7 + 12.7 + 12.7 + 12.7 9 in.
+ 9 + 9 = 68.6 in.
• Area: Note that the large
triangle can be moved to
make a rectangular figure.
– 9 • 18 = 162 in.2
18
in.
18 in.
Two
2m
• Perimeter:
– 10 + 10 + 2.8 + 2.8
+ 2.8 + 2.8 + 2 + 2 =
35.2 m
4 m2
14 m
• Area:
10 m
– Two trapezoids and a rectangle
– (.5)(2)(10 + 14) + (.5)(2)(10 + 14) + 2 • 14
– 84 m2
2.8 m
Volume of a Cube
• Take a block. Assume that each edge
measures 1 unit.
• Then, the volume of that block is
1 unit3.
• Use the blocks to make 2 other cubes.
How many unit3 are needed?
Surface Area of a Cube
• In a cube, all six faces are congruent.
• So, to find the surface area of a cube, we
simply need to find the area of one face, and
then multiply it by 6.
• If we don’t have a cube, but we have a
rectangular prism, there are still 6 faces: but
they are not all congruent.
• Front and back, top and bottom, right and
left.
Make rectangular prisms
• Make 3 different rectangular prisms, each
with a base of 6 cubes.
• The base must be a rectangle. Why?
• The area of the base remains constant.
Why?
• The only thing that changes is the height.
Why?
• What is the volume (number of cubes) of
each prism? Is this related to the L, W, and
H? If so, how?
Dimensions of
Rectangular Prisms
• 3x2x1
• 3x2x3
3x2x2
3x2x4
Rectangular prisms
• Volume: Volume is defined as area of
the base multiplied by the height.
• Why do we say L • W • H for a
rectangular prism?
height
width
length