unit 3: area, perimeter, and similarity lesson 3.1: area

Unit 3: Area, Perimeter, and Similarity

Lesson 3.1: Area of Triangles and Parallelograms Lesson 11.1 from textbook

Objectives • Find the areas of squares, rectangles, parallelograms, and triangles using area formulas.

• Find the dimensions of the triangle or quadrilateral with the given area.

Area of a Square Postulate

Area Congruence Postulate

Area Addition Postulate

Rectangle Area Theorem The area of a rectangle is ____________________________________________________ A = _______________

PARALLELOGRAMS Base = __________________________________ Height = _________________________________

Question: How do you find the area of a parallelogram? If we stretch rectangle ABCD into parallelogram ABCD………. Does the height or base length change? ______________ (Measure to find out, nearest mm) Does the area of the rectangle change? ______________ Critical Thinking:

How does the area of a triangle relate to the area of a rectangle?

A D

C B

A

B C

D

Parallelogram Area Theorem Triangle Area Theorem The area of a parallelogram is The area of a triangle is _________________________________ _____________________________ Example 1

A = ______ P = ________ A = ______ P = ________ A = ______ P = ________ Example 2 Find the area of each polygon

A = _______________ A = _______________

Example 3

B = _______________________ H = __________________ Example 4

Find the area of the polygon. A = _________________ IV. Application

A) You are making a tabletop in the shape of a parallelogram to replace an old 24 inch by 15 inch rectangular one. You want the areas of the table tops to be equal. The base of the parallelogram is 20 inches. What should the height be?

________________________________

B) You can mow 10 square yards of grass in one minute. How long does it take you to mow a

triangular plot with height 25 yards and base 24 yards? How long does it take you to mow a rectangular plot with base 24 yards and height 26 yards?

________________________ _________________________

5 in

4 in


Lesson 3.2: Areas of Trapezoids, Rhombuses, and Kites Lesson 11.2 from textbook

Objective • Find the areas of trapezoids, rhombuses, and kites using area formulas.

TRAPEZOID

Height of a trapezoid ___________________________

Bases of a trapezoid ___________________________

Trapezoid Area Theorem The area of the trapezoid is ________________________________. Example 1

Rhombus Area Theorem Kite Area Theorem The area of a rhombus is _____________________. The area of a kite is _________________.

Example 2 Find the area of each figure.

Example 3 Use the given information to find x.

Example 4 Find the area of the figure.

Example 5 The windshield in a truck is in the shape of a trapezoid. The lengths of the bases of the trapezoid are 70 inches and 79 inches. The height is 35 inches. Find the area of the glass in the windshield Example 6


Lesson 3.3: Find Arc Measures Lesson 10.2 from textbook

Objectives • Formally define and explain the different angles and arcs in a circle: central angle along with its

corresponding major arc, minor arc, or semicircle.

• Use properties of arcs to find their measure.

• Determine the measure of central angles and their associated major and minor arcs.

NAMING ARCS Minor Arcs _____________________________________ Major Arcs _____________________________________

MEASURING ARCS The measure of the minor arc is equal to _____________________________________________ The measure of the major arc is _________________________ ____________________________________________________ The measure of a semicircle is _________________________ Example 1

Find the measure of each arc of circle P, where RT is the diameter.

Arc Addition Postulate The measure of an arc formed by two adjacent arcs is ____________________________________________________ _____________________

Example 2 A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. Example 3 _______________________ _______________________ Example 4

Tell which if any of the given arcs are congruent. Explain why or why not.

____________________ _____________________ _____________________ Example 5

A clock with hour and minutes hands is set to 1:00 p.m. a) After 20 minutes, what will be the measure of the minor arc formed ____________________ by the hour and minute hands? b) At what time before 2:00p.m., to the nearest minute, will the hour and minute __________________ hands form a diameter?


Lesson 3.4: Circumference and Arc Length Lesson 11.4 from textbook

Objective • Find the circumference of a circle and the length of a circular arc using derived formulas.

Directions: In your teams, use your ruler and measure the diameter and circumference of each disc.

Record your results and then complete the table and answer the questions below.

Disc One Disc Two Disc Three

Diameter

Circumference

Circumference ÷ Diameter

1. What do you notice when you divide the diameter into the circumference for each disc?

__________________________________________________________________

2. Instead of measuring the circumference with a ruler, how would you calculate the circumference if

you only know the diameter? ___________________________________________________________________

3. If you only new the radius, what would the circumference be?

___________________________________________________________________

CIRCLES Circumference of a Circle

Name of circle ______________ ____________________________ Example 1 Find the exact circumference. Find the approximate radius.

C = _________ r ≈ ____________

Example 2 The dimensions of a car tire are shown at right. To the nearest Foot, how far does the tire travel in 15 revolutions? Question: How do we find the length of AB ? __________________________________ ARC OF A CIRCLE

AB is an arc on the circle

Arc Length Corallary

L = ____________________________

Example 3 Example 4 Find the length of the arc. Find the indicated measure.

__________________ _____________________ ___________________

Example 5 Example 6

Find the perimeter of the region.

P = _______________________

A measuring wheel is used to calculate the length of a path. The diameter of the wheel is 8 inches. The wheel rotates 87 times along the length of the path. About how long is the path?


Lesson 3.5: Areas of Circles and Sectors Lesson 11.5 from textbook

Objective • Find the area of a circle and the area of a sector of a circle using derived formulas.

• Find the area of a shaded region that is created by a composition of a polygon and a circle.

Area of a Circle Theorem Example 1

A = _________________ Find approximate area of a circle with radius 6 cm. A ≈ ________________ Example 2 Find the diameter of the circle with an area of 54cm2. d ≈ __________________

SECTORS

Question: How do you find the area of a sector?

Area of a Sector Theorem

P

A

B r

A sector of a circle is the region bounded by the two radii of the circle and the intercepted arc. Name of the sector _______________________ Bounded by _____________________________

A = _A = ______________________________

Example 3 Find the area of the sectors formed by <UTV. sector UV ≈ _______________

sector USV ≈ ______________

A

B

P

r

U

V

T

S

70o

8 cm

Example 4

Example 6 Find the length of the radius of the circle whose sector area is 50 cm2 and has an arc measure of 85o. r ≈ ___________________

Example 7 Find the area of the shaded region. A ≈ ____________________

Example 5

Use the diagram to find the area of circle S. A ≈ ___________________

S

F

9m

3.5 m

G

120o A = 214.37 cm2


Lesson 3.6: Ratios, Proportions, and the Geometric Mean Lesson 6.1 from textbook

Objectives • Find and simplify the ratio of two numbers.

• Use ratios of corresponding sides of two-dimensional figures to find the ratio of the perimeter and unknown side lengths.

• Use the cross product property to solve proportions. *Ratios are often expressed in their simplest form. Two ratios that have the same simplified form are called equivalent ratios. Example 1 You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484 feet and that the ratio of its length to its width is 9:2. Find the dimensions of the wall and find the area of the rectangle. Length ___________ Width ______________ Area ______________ Example 2 The measures of the angles in CDE∆ are in the extended ratio of 1:2:3. Find the measures of the angles. m<C = ____________ m<D = _____________ m<E = ___________

*Proportions: equation stating two ratios are equal. d

c

b

a= b and c are the means

a and d are the extremes Cross Products Property

1) The product of the means equals the product If d

c

b

a= where b ≠ 0 and c ≠ 0, then ______________

of the extremes.

Example 3 Solve the proportion.

xx 3

4

3

1=

− __________________

E D

C

Example 4 As part of an environmental study, you need to estimate the number of trees in a 150 acre area. You count 270 trees in a 2 acre area and you notice that the trees seem to be evenly distributed. Estimate the total number of trees. Example 5

Emily took 500 U.S. dollars to the bank to exchange for Canadian dollars. The exchange rate was 1.2 Canadian dollars per U.S. dollar. How many Canadian dollars did she get in exchange for the 500 U.S. dollars?

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x that satisfies b

x

x

a= .

So, x2 = __________________ and x = ______________________. Example 6

Find the geometric mean of 3 and 7. __________________


Lesson 3.7: Use Proportions to Solve Geometry Problems Lesson 6.2 from textbook

Objectives • Use properties of proportions to solve geometric problems.

Properties of Proportions

1) If two ratios are equal, then their reciprocals If d

c

b

a= , then _____________

=.

are also equal

2) If you interchange the means of a proportion, If d

c

b

a= , then _____________

=.

then you form another true proportion.

3) In a proportion, if you add the value of each If d

c

b

a= , then _____________

=.

ratio’s denominator to its numerator, then you form another true proportion,

Example 1

Determine whether the given proportion is true.

_________________ _______________

Example 2

In the diagram, .ST

NP

RS

MN= Write three true proportions and then solve for x.

_______________ ________________ _______________ x = ___________ Example 3

In the diagram, .EC

BE

DA

BD= Find BA and BD.

BA = ___________ BD = ___________

4

8 N

P

M

S R

T

10

x

3

E

C A

D

B

18

6

x 12

Example 4

JH = ___________ JL = ___________

Scale drawing A drawing that is the same shape as the object it represents.

Scale Ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object.

Example 5

The blueprint shows a scale drawing of a cell phone. The length of the antenna on the blueprint is 5 cm. The actual length of the antenna is 2 cm. What is the scale of the blue print?

Example 6 Two landmarks are 130 miles from each other. The landmarks are 6.5 inches apart on a map. Find the scale of the map.

Example 7 The scale for the diagram of the rectangular field is 1 inch = 50 yards. If the field in the diagram is 2 in by 3 in, what is the area and perimeter of the field?

EX: One inch on a map = One mile


Lesson 3.8 Use Similar Polygons Lesson 6.3 from textbook

Objectives • Use the definition of similarity to determine whether two polygons are similar.

• Use proportions to solve for the length of sides of similar polygons.

SIMILAR POLYGONS Congruent Angles ______________________________________ Ratios of

corresponding sides _____________________________________

Example 1 List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality. Congruent Angles ______________________________________ Ratios of corresponding sides _________________________________ Example 2 Determine whether the polygons are similar. If they are Write a similarity statement and find the scale factor. Similar? ________________ Similarity Statement ___________________ Scale Factor _____________ Example 3

In the diagram, the polygons are similar. Find the value of x. x = _______________

Perimeters of Similar Polygons Theorem If two polygons are similar, then the ratio of the perimeters _______________________________________________ _______________________________________________. Example 4 Use the similarity statement to find the scale factor of the polygon on the left to the polygon on the right. Then find the perimeter of the polygon. Scale Factor _____________ Perimeter ______________

CORRESPONDING LENGTHS IN

SIMILAR POLYGONS

If two polygons are similar, then the ratio of any two EX) Midsegments of a triangle. corresponding lengths in the polygons is equal to the scale factor of the similar polygons. ______________________________________

Example 5

In the figure, ABC∆ ~ DEF∆ . Find the value of x. x = ______________________

=PQ

KL_________________________

A

B

C

D E M

N

O

P

Q


Lesson 3.9 Perimeter and Area of Similar Figures Lesson 11.3 from textbook

Objective • Compare perimeters and areas of similar two-dimensional figures.

• Use the ratio of lengths of similar two dimensional figures to calculate the ratio of their perimeters and areas.

Ratio of the sides __________________________ Ratio of the perimeters _____________________ Area of small rectangle _____________________ Area of big rectangle _______________________ Ratio of the areas _________________________

Question: How does the ratio of the areas relate to the ratio of the perimeters?

Answer: __________________________________________________________________________

Areas of Similar Polygons Theorem If two polygons are similar with the lengths of corresponding sides in the ratio a:b, then the ratio of their areas is _________________________. Ratio of the sides _______________ Ratio of the areas ______________________ Example 1

In the diagram, ABC∆ ~ DEF∆ . Find the indicated ratio. a) Ratio of the perimeters _______________________ b) Ratio of the areas ___________________________

2

3 12

8

Example 2 Corresponding lengths in similar figures are given. Find the unknown area. Ratio of the sides _________________. Ratio of the areas _________________ Area of small trapezoid _____________________

Example 3 Use the given area to find XY. Ratio of the areas ___________________ Ratio of the sides ___________________ Proportion ________________________ XY = _____________

Example 4 You are installing the same carpet in a bedroom and a den. The floors of the rooms are similar. The carpet for the bedroom costs $225. Carpet is sold by the square foot. How much does it cost the carpet the den? Cost of carpet for den. ___________________ Example 5 Two rectangular banners from this year’s music festival are shown. Organizers of next year’s festiveal want to design a new banner that will be similar to a banner with dimensions 4 ft by 8 ft. The length of the new banner with the longest side will be 9ft. Find the area of the new banner. A = __________________________

unit 3: area, perimeter, and similarity lesson 3.1: area

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