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NOTES Geom Chap 11 Areas of Plane Figures Notes Chapter 11: Areas of Plane Figures Unit 1: Areas of Polygons Section 1: Areas of Rectangles on your desk Postulate 17 The area of a square 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 rectangle 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. squares 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, Triangles, and Rhombuses on your desk Theorem 11.2 Theorem 11.3 The area of a parallelogram equals The area of a triangle 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 rhombus 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 circle of 11.8 radius 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 regular polygon 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 hexagon 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 angle 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 octagon d) a regular dodecagon 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: Circles, Similar Figures, and Geometric Probability Section 5: Circumferences 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 Circumference, 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.
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