Notes Chapter 11: of Plane Figures Unit 1: Areas of Polygons Section 1: Areas of on your desk Postulate 17 The of a is the square of the length of a side. (A=s2) 11.1 11.2 Postulate 18 Area Congruence Postulate 11.3 If two figures are congruent, then they have the same area. 11.4

Postulate 19 Area Addition Postulate 11.5 The area of a region is the sum of the areas of its non-overlapping parts. 11.6 S D C 11.7 R 11.8

A B P Q

Theorem 11-1 There area of a equals the product of its base and height. (A=bh)

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 1: Areas of Rectangles on your desk Examples True or False. If false, state a counterexample. 11.1 1. If two figures have the same area, then they must be congruent. 11.2 2. If two figures have the same perimeter, then they must have the same 11.3 area. 11.4 3. If two figures are congruent, then they must have the same area. 4. Every square is a rectangle. 11.5 5. Every rectangle is a square. 11.6 6. The base of a rectangle can be any side of the rectangle. 11.7 11.8 7-10 Complete the table.

Questions 7 8 9 10

base 12 m 9 cm y!2

height 3 m y

Area 54 cm2 Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 1: Areas of Rectangles on your desk Practice 1. Find the area and perimeter of a 5. The lengths of the sides of three 11.1 square with sides 5 cm long. are s, s+1, and s+2. If 11.2 their total area is 365cm2, find 11.3 2. The perimeter of a square is their total perimeter. 11.4 28cm. What is the area?

11.5 3. The area of a square is 64cm2. 11.6 What is the perimeter? 6. Find the area in terms of the 11.7 variable. 4y 2y 11.8 4. The lengths of a rectangle is 7cm 3y longer than its width. If the y 2y 3y perimeter is 54cm, find the area of the rectangle. 8y

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 2: Areas of Parallelograms, , and on your desk Theorem 11.2 Theorem 11.3 The area of a parallelogram equals The area of a equals the half 11.1 the product of a base and the height the product of a base and the height 11.2 to that base. (A=bh) to the base. 11.3 11.4

11.5 11.6 11.7

11.8 Theorem 11.4 The area of a equals half the product of its diagonals. Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 2: Areas of Parallelograms, Triangles, and Rhombuses on your desk Practice Find the areas of each figure 6 11.1 1. 2. 3. 11.2 4 4 5 5 3 3 11.3 60° 11.4 4 6 6

11.5 11.6 11.7 11.8 4. 5. 5 6. 2 5 4 13 5 5 5 4 5 2 12 5

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 2: Areas of Parallelograms, Triangles, and Rhombuses on your desk Practice 7. Given AB=DE=GH, what can you conclude about the three triangles? 11.1 Justify your answer. 11.2 11.3 C F I 11.4

11.5 11.6 11.7 A B D E G H 11.8 Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 2: Areas of Parallelograms, Triangles, and Rhombuses on your desk Practice 7. Given AB=DE=GH, what can you conclude about the three triangles? 11.1 Justify your answer. 11.2 11.3 C F I 11.4

11.5 11.6 11.7 A B D E G H 11.8

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 2: Areas of Parallelograms, Triangles, and Rhombuses on your desk Practice 8. Use the diagram on right Q P 11.1 a) Find the area of PQRS. 11.2 O 20 16 11.3 b) Find the area of PSR 11.4 c) Find the area of OSR R 30 S 12 11.5 11.6 d) What is the area of PSO? 11.7

11.8 e) What must be the area of POQ be? Why? What must be the area of OQR be?

g) State what you have show in parts (a)-(e) about the diagonals of a parallelogram divide the parallelogram. How can you prove it? Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 3: Area of Trapezoids on your desk Theorem 11.5 The area of a trapezoid equals half the product of the height and the sum of 11.1 the bases. 11.2 D C 11.3 11.4

11.5 11.6 A B 11.7 11.8 Examples 1. 7 2. 5 3. 13

10 14 5 6 7

13 13 12

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 3: Area of Trapezoids on your desk Practice 1. In trapezoid TRAP, segment RA is parallel to segment TP, m"T=60, 11.1 RA=10, TR=8, and TP=15. Find the area of trapezoid TRAP. 11.2 11.3 11.4 2. In isosceles trapezoid ABCD, the legs are 8 and the bases are 6 and 14. 11.5 Find the area. 11.6 11.7 11.8 3. A trapezoid has an area of 75cm2 and a height of 5cm. How long is the median? Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 3: Area of Trapezoids on your desk Practice cont’d 8 5. 6. 7. 12 11.1 45° 8 8 30 ° 30 ° 11.2 30 11.3 60° 11.4 4

11.5 11.6 11.7 8. An isosceles trapezoid with bases 12 and 16 is inscribed in a of 11.8 10. The center of the circle lies in the interior of the trapezoid. Find the area of the trapezoid.

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 4: Areas of Regular Polygons on your desk

11.1 11.2 11.3 11.4

11.5 11.6 Theorem 11.6 11.7 The area of a is equal to half the product of the apothem, a 11.8 distance between center of the polygon to its side, and the perimeter. Examples 1. Find the area of a regular with apothem 6. Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 4: Areas of Regular Polygons on your desk Practice 1. Find the measure of a central for each inscribed figure: 11.1 a) a square 11.2 11.3 11.4 b) a regular hexagon 11.5 11.6 11.7 11.8 c) a regular

d) a regular

Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 4: Areas of Regular Polygons on your desk Practice cont’d 2. Find the perimeter and the area of each figure. 11.1 a) a regular hexagon with apothem 3 11.2 11.3 11.4

11.5 b) a regular octagon with sides 5 and apothem 3 11.6 11.7 11.8

c) an equilateral triangle with apothem 8 Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 4: Areas of Regular Polygons on your desk Practice cont’d 2. Find the perimeter and the area of each figure. 11.1 c) a regular hexagon with side 12 11.2 11.3 11.4

11.5 d) a square with diagonal 10 11.6 11.7 11.8

3. Find the area of regular pentagon inscribed in a circle of radius 8.

Notes Chapter 11: Areas of Plane Figures Unit 2: , Similar Figures, and Geometric Probability Section 5: and Areas of Circles on your desk central angle = 360/n, #=180/n By plugging in the value of n, we # 11.1 get the following values 11.2 a r Number of 11.3 Sides of 11.4 Polygon Perimeter Area 4 5.66r 2.00r2 s 11.5 6 6.00r 2.60r2 11.6 8 6.12r 2.83r2 11.7 10 6.18r 2.93r2 11.8 20 6.26r 3.09r2

30 6.27r 3.12r2

100 6.28r 3.14r2 Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 5: Circumferences and Areas of Circles on your desk In conclusion,

11.1 11.2 , C, of circle with radius r: C=2$r 11.3 Circumference, C, of circle with diameter d: C=$d 11.4 Area, A, of circle with radius r: A=$r2

11.5 11.6 Practice 11.7 1. Find the areas and the circumference of a circle with: 11.8 a. r=2 b. r=5 c. d=5

Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 5: Circumferences and Areas of Circles on your desk Practice cont’d 2. Find the radius of the circle whose area is numerical twice as much as the 11.1 circumference. 11.2 11.3 11.4

11.5 11.6 11.7 3. If a bicycle tire has a diameter of 26 inches, how far does it travel in 1000 11.8 revolution to nearest inch? Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 5: Circumferences and Areas of Circles on your desk Practice cont’d 4. The diameter of the world’s largest optical telescope at the 11.1 Zelenchuksakaya Observatory in the USSR is 6m. Find the circumference 11.2 of the lens. Find the areas and the circumference of a circle with: 11.3 11.4

11.5 11.6 5. Find the area of each shaded region. 11.7 11.8 2r 4r 4 r

Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 6: Arc Lengths and Areas of Sectors A on your desk In general, if mAB= Length of AB= x° 11.1 Area of sector AOB= O 11.2 B 11.3

11.4 Examples 1. The radius of a circle is 3cm. Find (a) the lengths of the given arcs, and 11.5 (b) the areas of the sectors determined by the given arcs. 11.6 11.7 (a) 50° (b) 20° (c) 140° (d) 210° 11.8 Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 6: Arc Lengths and Areas of Sectors on your desk Examples Cont’d A 2. Find the area of the regions bounded by the 11.1 segment AB and the arc AB. O 120° 11.2 11.3 6 B 11.4

11.5 11.6 11.7 3. A circle has area 160$. If a sector of the circle has area 40 $, find the 11.8 measure of the arc of the sector.

Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 7: Ratios of Areas on your desk Theorem 9-13 If the scale factor of two similar figures is a:b, then (1) the ratio of the 11.1 perimeter is a:b. (2) the ration of the areas is a2:b2. 11.2

11.3 Example 1 11.4 Find the ratios of 2 perimeter and the ratios 4 11.5 of areas of two similar 4 11.6 rectangles. 11.7 8 11.8

Example 2 Given the scales, what is 7 the ratio of two map of 4 California? Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 7: Ratios of Areas on your desk Example 3 A medium pizza with 10” diameter costs $10, and a large pizza with 12” 11.1 diameter costs $12. Which is a better buy? 11.2 11.3

11.4 Area of Circle= 6”

11.5 Area per Dollar= 11.6 11.7 11.8 Area of Circle= 5”

Area per Dollar=

Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 7: Ratios of Areas on your desk Here are some visual aid to understand

11.1 11.2 11.3 11.4

11.5 11.6 11.7 11.8

44% more

Bottom Line: Looks can be deceiving. So, you need to use both the common sense AND the analytical skills!! Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 7: Ratios of Areas on your desk Example 4 Find the ratio of the areas of %ABC and %ADC. 11.1 (a) B (b) C (c) A 12 D 11.2 5 15 20 8 11.3 9 11.4 B 9 A 16 D B 10 C A 11.5 C 11.6 11.7

11.8 Example 5 Find the ratio of the triangles (a) I to II and (b) I and III. (a) (b) 7 I I 4 III I I II III II 8 5 15

Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 8: Geometric Probability on your desk Example 1 A bus comes every 45 minutes. It waits for 5 minutes. If I showed up to the 11.1 same bus stop at random, what is the probability that the bus is waiting? 11.2 11.3 11.4

11.5 Example 2 11.6 As a beginning archer, when I hit a target, 11.7 the likelihood of each spot getting hit is the 11.8 same as any other spot. The radii of each circle are 1, 2, and 3. What is the probability that I will hit the bull eye? Notes Chapter 11: Areas of Plane Figures Unit 2: Circles, Similar Figures, and Geometric Probability Section 8: Geometric Probability on your desk Practice 1 Suppose a point P on segment AG is picked at random. What is the 11.1 probability that P is on the segment CF? 11.2 11.3 A B C D E F G 11.4

11.5 11.6 Practice 2 11.7 At a carnival game, dishes are positioned on a table so that they do not 11.8 overlap. You win a prize if you throw a nickel that lands in a dish. If the areas of the table is 5m2 and the combined area of the dishes is 1m2, what is the probability that the nickel will not land in a dish?