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Chapter 10

Chapter 10 . Area. Area of Rectangles and Parallelograms. Area = bh Where b is the base and h is the height perpendicular to the base. A.k.a. altitude. Area of a triangle. A = ½ bh. Examples. Find the area of the following to the nearest tenth. 1) 2) 3) 4).

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Chapter 10

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  1. Chapter 10 Area

  2. Area of Rectangles and Parallelograms Area = bh Where b is the base and h is the height perpendicular to the base A.k.a. altitude

  3. Area of a triangle A = ½ bh

  4. Examples. Find the area of the following to the nearest tenth. 1) 2) 3) 4)

  5. Area of a trapezoid

  6. Example:

  7. Area of a Kite and Rhombus

  8. Examples: Find the area to the nearest tenth. 1) 2)

  9. Day 2 Area of Polygons

  10. Apothem is the perpendicular distance from the center to a side. (a)

  11. Radius: From the center of the polygon to a vertex of the polygon. Used to find the apothem

  12. Areas of Regular Polygons

  13. Areas of Regular Polygons Perimeter Apothem

  14. Theorem 10-6: Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. A= ½ ap a = apothem p = perimeter

  15. Examples: Find the area to the nearest tenth. 1) 2)

  16. Example 1 Find the area of the octagon 9 in 7 in

  17. Examples • Find the area of each of the following: 11 yd 9 yd 12 cm 4 in 11 cm 9 in

  18. Using Special Right triangles • Use the properties of special right triangles to find the area of the hexagon. 10 in

  19. More Examples • Using properties of special right triangles, find the area of each hexagon. 6 3 m 12 yd

  20. Trigonometry and Area Find the area of the regular pentagon with 8cm sides. A = ½ (a)(40) 8 cm To find a: • Find the measure of < XCZ: • 360/5 = 72 C • Find the measure of < XCY: • 72/2 = 36 36 3. Use the triangle and trig to solve for a: a o = tan 36 = 4 a a a = 4 tan 36 a = 5.51 Use a to solve for area: A = ½ (5.51)(40) X 4 cm Z Y Area: 110.2 cm2

  21. Find the area of a regular octagon with a perimeter of 80 in. Give the area to the nearest tenth. To find the angle measure: 360/8 = 45 45/2 = 22.5 10 in o = tan 22.5 = 5 a a a = 5 tan 22.5 a = 12.07 a Use formula: A = ½ (12.07)(80) 5 in A = 482.8 in2

  22. Theorem 10.8: • The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. 3 Options:

  23. Finding the Area of a Triangle Area of Triangle ABC = ½ bh Area = ½ bc(sin A) Area = ½ (52)(24)(sin 76) Area = 605.46 in2 B a 24 in h 76 A C 52 in

  24. The surveyed lengths of two adjacent sides of a triangular plot of land are 412 ft and 386 ft. The angle between the sides is 71. Find the area of the plot. A = ½ bc(sin A) A = ½ (412)(386)(sin 71) 412 ft A = 75183.86 ft2 71 386 ft

  25. Two sides of a triangular building plot are 120 ft and 85ft long. They include an angle of 85. Find the area of the building plot to the nearest square foot. A = ½ bc(sin A) 85 85 ft A = ½ (120)(85)(sin 85) 120 ft A = 5081 ft2

  26. Warm Up • Find the area of a regular triangle with an apothem of 4”. • Find the area of a decagon with side lengths of 15 cm.

  27. Using Trigonometry

  28. Example

  29. Circles and Arcs Radius Diameter Center

  30. Central Angle Q R S An angle whose vertex is the center of the circle.

  31. Arcs • Semicircle: half of the circle • Minor Arc: smaller than a semicircle • Major Arc: Greater than a semicircle

  32. Example • Name three minor arcs, two semicircles, and three major arcs.

  33. Adjacent Arcs • Arcs of the same circle that have exactly one point in common • Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs

  34. Example • Find the measure of the following arcs 1. 2. 3. 4.

  35. Concentric Circles • Circles in the same plane with the same center Not Concentric Concentric

  36. Circumference In terms of pi Nearest tenth

  37. Arc Length • The length of the fraction of the circle’s circumference

  38. Area In terms of pi Nearest tenth

  39. Sector • A region bounded by an arc of the circle and two radii

  40. Area of a Sector • The product of the ratio of the sector and the area

  41. Examples: Find the area of the sector shaded.

  42. Segment • A part of a circle bounded by an arc and the segment joining its endpoints

  43. Area of the segment:

  44. Examples:

  45. Find the area of the shaded region:

  46. Geometric Probability

  47. Find the probability of getting a dart in the center of this triangular board:

  48. Find the probability of hitting one of the circles in the box with a dart:

  49. 12-1 Tangent Lines

  50. 12-1: Tangent Lines • Tangent to a circle: a line in the plane of the circle that intersects the circle in exactly one point. Point of Tangency

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