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9.2 – Curves, Polygons, and Circles

9.2 – Curves, Polygons, and Circles. Curves. The basic undefined term curve is used for describing non-linear figures in a plane. A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice.

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9.2 – Curves, Polygons, and Circles

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  1. 9.2 – Curves, Polygons, and Circles Curves The basic undefined term curve is used for describing non-linear figures in a plane. A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice. A closed curve has the same starting and ending points, and is also drawn without lifting the pencil from the paper. Simple; closed Simple; not closed Not simple; closed Not simple; not closed

  2. 9.2 – Curves, Polygons, and Circles Polygons A polygon is a simple, closed curve made up of only straight line segments. The line segments are called sides. The points at which the sides meet are called vertices. Polygons with all sides equal and all angles equal are regular polygons. Regular Polygons Polygons

  3. 9.2 – Curves, Polygons, and Circles A figure is said to be convex if, for any two points A and Binside the figure, the line segment AB is always completely inside the figure. E F B A M C N D Convex Not convex

  4. 9.2 – Curves, Polygons, and Circles Classification of Polygons According to Number of Sides Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Nonadecagon Icosagon Hendecagon Triacontagon Dodecagon Tetracontagon

  5. 9.2 – Curves, Polygons, and Circles Types of Triangles - Angles All Acute Angles One Right Angle One Obtuse Angle Acute Triangle Right Triangle Obtuse Triangle

  6. 9.2 – Curves, Polygons, and Circles Types of Triangles - Sides All Sides Equal Two Sides Equal No Sides Equal Equilateral Triangle Isosceles Triangle Scalene Triangle

  7. 9.2 – Curves, Polygons, and Circles Quadrilaterals: any simple and closed four-sided figure A trapezoid is a quadrilateral with one pair of parallel sides. A parallelogram is a quadrilateral with two pairs of parallel sides. A rectangle is a parallelogram with a right angle. A square is a rectangle with all sides having equal length. A rhombus is a parallelogram with all sides having equal length.

  8. 9.2 – Curves, Polygons, and Circles Triangles Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180°. Find the measure of each angle in the triangle below. x° (220 – 3x)° (x+20)° x + x + 20 + 220 – 3x = 180 x = 60° –x + 240 = 180 60+ 20 = 80° – x = – 60 220 – 3(60) = 40° x = 60

  9. 9.2 – Curves, Polygons, and Circles Exterior Angle Measure The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. 2 Exterior angle 4 1 3 The measure of angle 4 is equal to the sum of the measures of angles 2 and 3. m4 = m2 + m3

  10. 9.2 – Curves, Polygons, and Circles Find the measure of the exterior indicated below. (x+20)° (3x – 50)° (x – 50)° x° 3x – 50 = x + x + 20 3x – 50 = 2x + 20 3x = 2x + 70 x = 70 3(70) – 50 160°

  11. 9.2 – Curves, Polygons, and Circles Circles A circle is a set of points in a plane, each of which is the same distance from a fixed point (called the center). A segment with an endpoint at the center and an endpoint on the circle is called a radius (plural: radii). A segment with endpoints on the circle is called a chord. A segment passing through the center, with endpoints on the circle, is called a diameter. A diameter divides a circle into two equal semicircles. A line that touches a circle in only one point is called a tangent to the circle. A line that intersects a circle in two points is called a secant line. A portion of the circumference of a circle between any two points on the circle is called an arc.

  12. 9.2 – Curves, Polygons, and Circles O is the center P PQ is an arc. M OQ is a radius. LM is a chord. Q O PQ is a secant line. PR is a diameter. (PQ is a chord). L T R RT is a tangent line.

  13. 9.2 – Curves, Polygons, and Circles Inscribed Angle An inscribed angle is an angle whose vertex is on the circle and the sides of the angle extend through the circle and touch or go beyond the points on the circle. BAC is an inscribed angle B x C A Any angle inscribed in a semicircle must be a right angle. B A C

  14. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Congruent Triangles Congruent triangles: Triangles that are both the same size and same shape. E B A D C F The corresponding sides are congruent. The corresponding angles have equal measures. Notation:

  15. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Using Congruence Properties Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent. B E A D C F

  16. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Using Congruence Properties Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent. E B A D C F

  17. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Congruence Properties - SSS Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent. B E A D C F

  18. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Proving Congruence CE = ED CE = ED Given: C B AE = EB AE = EB E A Prove: ACE  BDE D Given 1. 1. 2. 2. Given 3. AEC = BED 3. Vertical angles are equal ACE  BDE 4. 4. SAS property

  19. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Proving Congruence B Given: ADB = CBD C ABD = CDB A D Prove: ABD  CDB Given 1. ADB = CBD 1. 2. ABD = CDB 2. Given 3. BD = BD 3. Reflexive property 4. 4. ABD  CBD ASA property

  20. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Proving Congruence B AD = CD AD = CD Given: AB = CB AB = CB BD = BD Prove: ABD  CBD A C D Given 1. 1. 2. 2. Given 3. 3. Reflexive property 4. 4. ABD  CBD SSS property

  21. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Isosceles Triangles If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold. B 1. The base angles A and C are equal. 2. Angles ABD and CBD are equal. 3. Angles ADB and CDB are both right angles. A D C

  22. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Similar Triangles:Triangles that are exactly the same shape, but not necessarily the same size. For triangles to be similar, the following conditions must hold: 1. Corresponding angles must have the same measure. 2. The ratios of the corresponding sides must be constant. That is, the corresponding sides are proportional. Angle-Angle Similarity Property If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar.

  23. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem B E 8 16 F 24 C D 32 A Find the length of side DF. Set up a proportion with corresponding sides: DF = 16.

  24. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Pythagorean Theorem c (hypotenuse) leg a leg b If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then (The sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.)

  25. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Find the length a in the right triangle below. 39 a 36

  26. 9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Converse of the Pythagorean Theorem If the sides of lengths a, b, and c, where c is the length of the longest side, and if then the triangle is a right triangle. Is a triangle with sides of length 4, 7, and 8, a right triangle? Is a triangle with sides of length 8, 15, and 17, a right triangle? Not a right triangle. right triangle.

  27. 9.4 – Perimeter, Area, and Circumference Perimeter of a Polygon The perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units. Perimeter of a Triangle a b c The perimeter P of a triangle with sides of lengths a, b, and c is given by the formula: P = a + b + c.

  28. 9.4 – Perimeter, Area, and Circumference Perimeter of a Rectangle w l The perimeter P of a rectangle with length l and width w is given by the formula: P = 2l + 2w or P = 2(l + w). s Perimeter of a Square s The perimeter P of a square with all sides of length s is given by the formula: P = 4s.

  29. 9.4 – Perimeter, Area, and Circumference Area of a Polygon The amount of plane surface covered by a polygon is called its area. Area is measured in square units. l Area of a Rectangle w The area A of a rectangle with length l and width w is given by the formula: A = l w. s Area of a Square s The area A of a square with all sides of length s is given by the formula: P = s2.

  30. 9.4 – Perimeter, Area, and Circumference h Area of a Parallelogram b The area A of a parallelogram with height h and base b is given by the formula: A = bh. b1 Area of a Trapezoid h b2 The area A of a trapezoid with parallel bases b1and b2and height h is given by the formula: A = (1/2) h (b1 + b2)

  31. 9.4 – Perimeter, Area, and Circumference Area of a Triangle h b The area A of a triangle with base b and height h is given by the formula: A = (1/2) b h

  32. 9.4 – Perimeter, Area, and Circumference Find the perimeter and area of the rectangle. 15 ft 7 ft Perimeter P = 2l + 2w 2(15) + 2(7) P = 44 ft Area A = lw A = (15)(7) A = 105 ft2 Find the area of the trapezoid. 7 cm. 6 cm. A = (1/2) h (b1 + b2) 6 cm. 5 cm. A = (1/2) (5) (7 + 13) 13 cm. A = (1/2) (5) (20) A = 50 cm2

  33. 9.4 – Perimeter, Area, and Circumference Find the area of the shaded region. 4 in. Area of square – Area of triangle s2 – (1/2) bh 4 in. 42 – (1/2) (4)(4) 16 – 8 in2 8

  34. 9.4 – Perimeter, Area, and Circumference Circumference and Area of a Circle r d The circumference C of a circle of diameter d is given by the formula: wherer is a radius. or The area A of a circle with radius r is given by the formula:

  35. 9.4 – Perimeter, Area, and Circumference Find the area and circumference of a circle with a radius that is 6 inches long (use 3.14 as an approximation for ). Circumference (  3.14) Circumference () C = 2  r C = 2  r C = 2 (3.14) 6 C = 2  6 C = 37.68 in C = 37.699 in Area (  3.14) Area () A =  r2 A =  r2 A =  62 A = (3.14) 62 A = 113.04 in2 A = 113.097 in2

  36. 9.5 – Space Figures, Volume, and Surface Area Space figures: Figures requiring three dimensions to represent the figure. Polyhedra: Three dimensional figures whose faces are made only of polygons. Regular Polyhedra: A polyhedra whose faces are made only of regular polygons (all sides are equal and all angles are equal.

  37. 9.5 – Space Figures, Volume, and Surface Area Other Polyhedra Pyramids are made of triangular sides and a polygonal base. Prisms have two faces in parallel planes; these faces are congruent polygons.

  38. 9.5 – Space Figures, Volume, and Surface Area Other Space Figures Right Circular Cones have a circle as a base and the surface tapers to a point directly above the center of the base. Right Circular Cylinders have two circles as bases, parallel to each other and whose centers are directly above each other.

  39. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume is a measure of capacity of a space figure. It is always measured in cubic units. Surface Area is the total region bound by two dimensions. It is always measured in square units. Volume and Surface Area of a Rectangular solid (box) The volume V and surface area S of a box with length l, width w, and height h is given by the formulas: V = lwh h and w S = 2lw + 2lh + 2hw l

  40. 9.5 – Space Figures, Volume, and Surface Area Find the volume and surface area of the box below. 3 in. 2 in. 7 in. V = 7(2)(3) S = 2(7)(2) + 2(7)(3) + 2(3)(2) V = 42 in.3 S = 28 + 42 + 12 S = 82 in.2

  41. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume and Surface Area of a Cube The volume V and surface area S of a cube with side lengths of s are given by the formulas: V = s3 s and s S = 6s2 s Find the volume and surface area of the cube below. S = 6(5)2 V = 53 V = 125 ft.3 S = 625 S = 150 ft.2 5 ft.

  42. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Right Circular Cylinder The volume V and surface area S of a right circular cylinder with base radius r and height h are given by the formulas: V = r2h and h r S = 2rh + 2r2 Find the volume and surface area of the cylinder below. V = (2)2(10) S = 2(2)(10) + 2(2)2 V = 40 S = 40 + 8 = 48 10 m 2 m V = 125.6 m3 m2 S = 150.72

  43. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Sphere The volume V and surface area S of a sphere radius r are given by the formulas: r V = (4/3) r3 and S = 4 r2 Find the volume and surface area of the sphere below. V = (4/3)(9)3 S = 4 (9)2 9 in. V = 972 S = 324 V = 3052.08 in.3 S = 1017.36 in.2

  44. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Right Circular Cone The volume V and surface area S of a right circular cone with base radius r and height h are given by the formulas:

  45. 9.5 – Space Figures, Volume, and Surface Area Find the volume and surface area of the cone below. V = (1/3)(3)2(4) V = 12 V = 37.68 m3 h = 4 m r = 3 m S = 15 + 9 S = 24 S = 75.36 m2

  46. 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of a Pyramid The volume V of a pyramid with height h and base of area B is given by the formula: Note: B represents the area of the base (l w). Find the volume of the pyramid (rectangular base) below. 7 cm 3 cm cm3 6 cm

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