GEOMETRY

CHAPTER 3

Geometry & Measurement

3.1 Measuring Distance, Area and Volume

3.2 Applications and Problem Solving

3.3 Lines, Angles and Triangles

3.1 Rounding Measurements

To round: 1. Underline the place 2. If number to the right of the under-lined place is 5 or more, add one 3. Otherwise, do not change4. Change all digits to the right of underlined number to zeros

3.1 Rounding Example

1. 38.67

2. First number to the right of 8 is “6”, so add one to 8

4. Change all digits to the right to 0’s. The answer is, 39.00 or 39

Example: Round 38.67 centimeters to the nearest centimeter

3.1 Calculating Distances

Linear Measure - a distance which could be around a polygon (perimeter) or around a circle (circumference)

Perimeter - sum of the lengths of the sides

C d r= =π (d Re member )2Circumference - distance around circle

Measure can be in U.S. system (yd, ft, etc.) or metric (cm,m, etc)

3.1 Metric Measures

King

Milk

Henry Died

Drinking

Monday

Chocolate

km hm dam m

dm cm mm

kilometer hectometer dekameter meter

decimeter centimeter millimeter

3.1 Metric Measures

1 cm = 0.01 m

1 dm = 0.1 m

1 mm = 0.001 m

1 hm = 100 m

1 km = 1000 m

1 dam = 10 m

3.1 Linear Distance

2. What is the distance around the polygon, in meters?

75 cm78 cm

95 cm 80 cm 78 + 95 +

80 + 75 = 328 cm

km

A. 328 m

hm dam m dm cm mm

B. 32.8 m

C. 3.28 m D. 0.328m

3.1 Calculating Areas

Rectangle

Parallelogram

Square

Triangle

Trapezoid

Circleb1

b2

r

3.1 Area - Square Units

4. What is the area of a circular region whose diameter is 6 cm?

If d = 6,

then r = 3A = r 2π

Formula:

=π (3)2

€

D. 9π sq. cm

€

B. 6π sq. cm

€

A. 36π sq. cm

€

C. 12π sq. cm

Surface area of a rectangular solid

3.1 Examples of Area

LW

H

There are 6 faces of the solid

A=2LH

Front/backSides (Left/Right)

+2WH

Top/Bottom

+2LW Square units

3.1 Examples of Area

6. What is the surface area of a rectangular solid that is 12 in. long, 5 in. wide and 6 in. high?

D. 360 sq. in.

L=12W=5

H=6

A=2(12)(6)+2(5)(6)+2(12)(5)A=2LH +2WH +2LW

A. 360 cubic in. B. 324 sq. in.

C. 324 cubic in.

3.1 Volume - Cubic Units

Rectangular Solid

Cylinder

h

h

h Cone

Sphere

V=lwh

V r h=π 2

hrV 2 3

1π=

V r=43

3π

r

rr

w l

3.1 Example of Volume

8. What is the volume of a sphere with a 12 inch diameter?

Formula: V r=43

3π If d = 12,then r = 6

V =43

6 3π ( ) Since (6)(6)(6)= 216, the only reasonable ans. is C

3.1 Identifying the Unit

9. Which of the following would not be used to measure the amount of water needed to fill a swimming pool?

A. Cubic feet

Think of “volume” as capacity or filling up the inside of a 3D figure.

linear

B. Liters

C. Gallons D. Meters

3.2 Application Example

1. What will be the cost of tiling a room measuring 12 ft. by 15 ft. if square tiles cost $2 each & measure 12 in.?

Since 12 inches = 1 ft, one tile is 1 ft on each side or 1 sq. ft.Area room: A = bh; (12)(15) = 180 sq ft And (180)($2) = $360 cost

A. $180 B. $4320 C. $360 D. $3600

3.2 Pythagorean Theorem

For any RIGHT TRIANGLE c

c a b2 2 2= +

a

b

Side opposite the right angle is the hypotenuse “c”

3.2 Pythagorean Theorem

c a b2 2 2= +

3. A TV antenna 12 ft. high is to be anchored by 3 wires each attached to the top of antenna and to pts on the roof 5 ft. from base of the antenna. If wire costs $.75 per ft, what will be the cost?

12

€

c2 = (12)2 +(5)2 = 144 + 25 = 169

Cost is .75 x 39 =$29.25

5

c

€

c = 13 and 3 wires x 13 ft = 39 ft

A. $27.00 B. $29.25 C. $9.75 D. $38.25

3.2 Infer & Select Formulas

7. The figure shows a regular hexagon Select the formula for total area

b

Total area is the area of the 6 identical triangles.

A. 3h+b

If area of 1 triangle = 1/2xbh,then 6 x 1/2 x bh = 3 bh

B. 6(h+b) C. 6hb D. 3hb

h

3.3 Lines; Angles; Triangles

straight angle 180right angle 90obtuse > 90, < 180acute angle < 90comp. sum to 90supp. sum to 180vertical angles-equal

ANGLES TRIANGLES

Right triangle Acute triangle Obtuse triangle Scalene triangle Isosceles Equilateral

3.3 Properties Example

2. What type of triangle is ABC?

55°

70°

A. Isosceles

Since sum of angles of triangle = 180,and 55 + 70 = 125,then angle C = 180 - 125 = 55.If 2 angles = , then isosceles. C

B. RightC. Equilateral D. Scalene

3.3 Angle Measures

BB

BB

SS

SS

1. B S Theorem All B’s are = , All S’s are = B + S = 180

2. Perpendicular lines intersect to form right angles.

3.3 Angle Measures

7

L11Terminology

The parallel lines are cut by transversal T

L2

45

8

32

6

Corresponding angles are =1 and 5, 3 and 7, 2 and 6, 4 and 8

Vertical angles are =1 and 4, 3 and 2, 6 and 7, 5 and 8

T

3.3 Angle Measures

7

L11Terminology

The parallel lines are cut by transversal T

L2

45

8

32

6

TAlternate interior angles are =4 and 5, 3 and 6

Alternate exterior angles are =1 and 8, 2 and 7

3.3 Angle Measures

3. If 2 angles of a triangle are = , then sides opposite are =

4. If 2 sides of a triangle are =, then angles opposite are =

3.3 Examples

7575

75

75

10545

45

4545

135

6. Which statement is true for the figure shown at the right given that L1 and L2 are parallel?

After using the BS theorem, angle T does = 75 and angle S=105

€

A.Sincem∠T = 75°,m∠S = 60°

TSV

R

€

B.Sincem∠T = 75°,m∠S = 105°

€

C.m∠V = m∠R

€

D.None

L1

L2

135105

60

60

3.3 Similar Triangles

Two triangles are similar if all angles are = and sides proportional

10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD

A. i only B. ii only C. i and ii only D. i, ii, iii

4040

7.5A

x

E5

CD

B4

Since m D=m B and DCE and ACB areVertical angles m A=m E

3.3 Similar Triangles

Two triangles are similar if all angles are = and sides proportional

10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD

A. i only B. ii only C. i and ii only D. i, ii, iii

4040

7.5A

x

E5

CD

B4

The triangle are similar, thus ratios of corresponding sides are =. x/4 = 7.5/5 thus x= 4(7.5)/5 = 6

3.3 Similar Triangles

Two triangles are similar if all angles are = and sides proportional

10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD

A. i only

4040

7.5A

x

E5

CD

B4

The triangle are similar, thus ratios of corresponding sides are =. CE/CA = CD/CB thus iii is false!

B. ii only C. i and ii only D. i, ii, iii

REMEMBER

MATH IS FUN

AND …

YOU CAN DO IT