Math 213 Student Note Outlines

Sections

9.2 9.3 11.1 11.2 11.3

§9.2 KEY IDEAS, page 1 of 2 Polygon Vertex Angles Sum Regular Polygons: Vertex Angles Congruence Definition

Examples

Regular Polygons Definition

Examples

Tessellation Definition Triangles? Quadrilaterals? Convex or concave? Pentagons? Hexagons?

§9.2 KEY IDEAS, page 2 of 2 Regular Tessellations Definition

Examples

Semi-Regular Tessellations Definition

Examples

§9.3 KEY IDEAS, page 1 of 3 Polyhedron (polyhedra) Definition Polyhedron Definition Polyhedron Edge

Definition Polyhedron Face

Definition Polyhedron Vertex

Definition Solid Definition Convex Polyhedron Definition Concave Polyhedron Definition Regular Polyhedron Platonic Solids Cube (hexahedron)

Tetrahedron

Octahedron Dodecahedron Icosahedron Definition Semi-Regular Polyhedron

§9.3 KEY IDEAS, page 2 of 3 Pyramids and Prisms Definition Pyramid Examples Example D Definition Prism Examples Cones and Cylinders Definition Cone Examples Definition Cylinder Examples

§9.3 KEY IDEAS, page 3 of 3 Spheres Definition Sphere Meridians of Latitude Definition Examples Meridians of Longitude Definition Examples Problem #19 Two points on the Earth’s surface that are on opposite ends of a line segment through the center of the Earth are called antipodal points. The coordinates of such points are nicely related. The latitude of one point is as far above the equator as that of the other is below, and the longitudes are supplementary angles (in opposite hemispheres). For example, (308N,

158W) is off the west coast of Africa near the Canary Islands, and its antipodal point (308S, 1658E) is off the eastern coast of Australia. a. This globe shows that (208N, 1208W) is a point in the Pacific Ocean just west of Mexico. Its antipodal point is just east of Madagascar. What are the coordinates of this antipodal point? b. The point (308S, 808E) is in the Indian Ocean. What are the coordinates of its antipodal point? In what country is it located?

§11.1 KEY IDEAS, page 1 of 2 Mappings Congruent Polygons Corresponding Sides Corresponding Angles Examples Triangle Congruence Properties Side – Side – Side (SSS) Examples Side – Angle – Side (SAS) Examples

§11.1 KEY IDEAS, page 2 of 2 Angle – Side – Angle (ASA) Examples SSA: Not a property Examples

§11.2 KEY IDEAS, page 1 of 1 Translations Example A Reflections Lines of Reflection Examples Rotations Points of Rotation Examples Compositions of Mappings Glide Reflection Examples Example D

§11.3 KEY IDEAS, page 1 of 2 Similarity and Scale Factors Example A Similar Figures: Definition Similar Polygons Definition Examples B & C Similar Triangles Definition Example D Angle-Angle Similarity Property Example E

§11.3 KEY IDEAS, page 2 of 2 Side-Side-Side Similarity Property Examples G Scale Factors—Surface Area & Volume Example H—Surface Area Example I—Volume

Math 213 Class Activity Handouts

9.2 Drawing Regular Polygons

10.2 Problem Solving

10.3 Problem Solving

11.1 Triangle Activity

11.2 Problem Solving

11.3 Problem Solving

Section 9.2: Drawing Regular Polygons Math 213 BBN: Conceptual Approach, page 594 Carefully follow the directions and do each of the following:

1. Draw a REGULAR PENTAGON using the Vertex Angle technique. 2. Draw a REGULAR HEXAGON using the Inscribed Polygon in a Circle technique 3. Draw a REGULAR HEPTAGON using either the Vertex Angle technique or the Inscribed

Polygon in a Circle technique (use the back of this sheet).

Math 213, Section 10.2 Problem Solving 1. Explain how to get the area formula for a parallelogram in a way that a child would

understand. 2. Explain how to get the area formula for a trapezoid in a way that a child would understand. 3. 10.2 Problem Opener

a. Original: Each of the 10 equilateral triangles in the following figure has sides of length 1 unit, and the perimeter of the entire figure is 12 units. What will the perimeter of the figure be if it is extended to include 50 such triangles?

b. Extension 1: Each of the 10 squares in the following figure has sides of length 1 unit, and

the perimeter of the entire figure is ______ units. What will the perimeter of the figure be if it is extended to include 50 squares?

c. Extension 2: Each of the 10 pentagons in the following figure has sides of length 1 unit, and the perimeter of the entire figure is ______ units. What will the perimeter of the figure be if it is extended to include 50 pentagons?

Math 213, Section 10.2 Problem Solving 4. Explain how to get the area formula for a parallelogram in a way that a child would

understand.

5. Explain how to get the area formula for a trapezoid in a way that a child would understand.

6. Section 10.2 Questions #30, 32, 34 and 36. #30 A pane of antique stained glass has dimensions of 30 centimeters x 58 centimeters. If the glass sells for 25 cents per square centimeter, what is the cost of the pane? #32 All-purpose carpeting costs $27.50 per square meter. What is the cost of carpeting a room from wall to wall whose dimensions are 360 centimeters x 400 centimeters? #34 The length of a kitchen cupboard is 3.5 meters. There are three shelves in the cupboard, each 30 centimeters wide. How many rolls of shelf paper will be needed to cover these shelves if each roll is 30 centimeters x 3 meters? #36 Some humidity is necessary in homes for comfort, but too much can cause mold and peeling paint. Paint destroying moisture can come from walls, crawl spaces, and attics.

a. An attic should have 900 square

centimeters of ventilation for each 27 square meters of floor area. How many square centimeters of ventilation are needed for an attic with a 4-meter x 12-meter floor?

b. A crawl space should have 900 square

centimeters of ventilation for each 27 square meters of ceiling area, plus 1800 square centimeters for each 30 meters of perimeter around the crawl space. How many square centimeters of ventilation are needed for a 10-meter x 15-meter crawl space?

213: Problem Solving (10.3) 1. (AA 10.3 #8) Spherical objects such as Christmas tree ornaments and basketballs

are packed in cube shaped compartments and boxes.

a. If a sphere just fits into a box, estimate what fraction of the box is “wasted

space.” Circle the fraction that best represents your estimate.

Compute the amount of wasted space when a sphere of diameter 25 centimeters is placed in a 25 cm × 25 cm × 25 cm box. What fraction of the total volume of this box is wasted space?

b. Spheres are also packed in cylinders. A cylinder provides a better fit than a box, but there is still extra space. If 3 balls are packed in a cylindrical can whose diameter equals that of a ball and whose height is 3 times the diameter of a ball, estimate what fraction of the space is unused.

Compute the amount of unused space. Assume that each ball and the cylinder have radius r and then express the height of the can in terms of r.

c. Tennis ball cans hold 3 balls. Determine the amount of wasted space in these cans if the balls have a diameter of 6.3 centimeters and the can has a diameter of 7 centimeters and a height of 20 centimeters. What fraction from part b is closest to representing the amount of wasted space? Check the reasonableness of your answer by experimenting with a can of balls and water.

2. Volume and Surface Area for Prisms and Pyramids

a. Regular octagon What is the area of this regular octagon?

b. Right regular octagonal prism: The bases are the same size as the octagon in

part a.

What is the surface area of the right regular octagonal prism? What is the volume of the right regular octagonal prism?

c. Right regular octagonal pyramid: The base is the same size as the octagon in part a. and the pyramid is the same height as the prism in part b.

What is the surface area of the right regular octagonal pyramid? What is the volume of the right regular octagonal pyramid?

11.1 Triangle Activity Math 213 For the ten triangles on the next page: Carefully decide which triangles are congruent. Angles that look like right angles are right angles. a. For each congruent pair, state the congruence relationships such as CDAB ≅ or

BBBAAA "''" ∠≅∠ . b. For each congruent pair, explain which congruence property (SSS, SAS, ASA)

determines the congruence.

1

m∠STU = 90.00°m TU = 3.00 cm

m US = 5.00 cm

U T

S

6 m∠CAT = 90.00°

m∠TCA = 45.00°m AC = 3.00 cm

T

A C

2 m∠ZXY = 90.00°

m ZX = 3.00 cm

m XY = 4.00 cm

Z

X

7

m LN = 3.00 cm

m∠MNL = 70.00°

m∠NML = 55.00°

NL

M

3

BC = 3.00 cm

m∠ABC = 115.00°

m CA = 5.06 cm

BA

C

8

CD = 3.00 cm

m∠DCE = 115.00°

m ED = 5.06 cm

DC

E

4 m∠DOG = 55.00°

m DO = 3.00 cm

m∠ODG = 70.00°OD

G

9

EF = 3.00 cm

DE = 3.00 cm

m∠DFE = 30.00°

ED

F

5

m∠PRS = 90.00°

m PR = 3.00 cm

m RS = 3.00 cmSR

P

10

VU = 3.00 cm

WV = 3.00 cm

m∠VUW = 30.00°

VW

U

11.2 Problem Solving Page 1 of 3

11.2 Problem Solving Page 2 of 3

11.2 Problem Solving Page 3 of 3

11.3 PROBLEM SOLVING Math 213

1. Draw a pentagon, P1, on grid paper a. What is the perimeter of pentagon P1? b. What is the area of pentagon P1? c. Sketch a new pentagon, P2, which is P1 scaled by 2. d. What is the perimeter of pentagon P2? e. What is the area of pentagon DP2? f. Sketch a new pentagon, P½, which is P1 scaled by ½. g. What is the perimeter of pentagon P½? h. What is the area of pentagon P½?

Do text problem 11.3 #10

2. Determine whether the triangles in each pair in exercises 9 and 10 are similar. If so, give a reason and write the similarity correspondence.

a. b.

c.

Do text problem 11.3 #31

3. During a football game, Beth and her friend were standing next to the goal posts and trying to estimate the height of the posts. Beth noticed that the length of the shadow of the posts was 10 yards, the length of the end zone, and her friend’s shadow had a length of 6 feet. Knowing that her friend was 6 feet tall, she quickly computed the height of the goal posts.

a. What is the height of the posts in feet? Explain your reasoning. b. In 1989, a change in football regulations allowed the height of the goal posts to be

increased by 10 feet. If the posts Beth saw had been 10 feet higher, what would have been the length of their shadow in yards?

Section 9.2: Drawing Regular Polygons Math 213BBN: Conceptual Approach, page 594Carefully follow the directions and do each of the following: