17 GEOMETRY.ppt
- 1. INTRODUCTION TO GEOMETRY MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur
- 2. Geometry • The word geometry comes from Greek words meaning “to measure the Earth” • Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics
- 3. The Greeks and Euclid • Our modern understanding of geometry began with the Greeks over 2000 years ago. • The Greeks felt the need to go beyond merely knowing certain facts to being able to prove why they were true. • Around 350 B.C., Euclid of Alexandria wrote The Elements, in which he recorded systematically all that was known about Geometry at that time.
- 4. The Elements • Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms: Point (Straight) Line Plane (Surface) Actually, Euclid did attempt to define these basic terms . . . is that which has no part is a line that lies evenly with the points on itself is a plane that lies evenly with the straight lines on itself
- 5. Basic Terms & Definitions • A ray starts at a point (called the endpoint) and extends indefinitely in one direction. • A line segment is part of a line and has two endpoints. A B AB B A AB
- 6. • An angle is formed by two rays with the same endpoint. • An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees. vertex side side
- 7. • A right angle has a measure of 90 degrees. • A straight angle has a measure of 180 degrees.
- 8. • A simple closed curve is a curve that we can trace without going over any point more than once while beginning and ending at the same point. • A polygon is a simple closed curve composed of at least three line segments, called sides. The point at which two sides meet is called a vertex. • A regular polygon is a polygon with sides of equal length.
- 9. Polygons # of sides name of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon
- 10. Quadrilaterals • Recall: a quadrilateral is a 4-sided polygon. We can further classify quadrilaterals: A trapezoid is a quadrilateral with at least one pair of parallel sides. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A kite is a quadrilateral in which two pairs of adjacent sides are congruent. A rhombus is a quadrilateral in which all sides are congruent. A rectangle is a quadrilateral in which all angles are congruent (90 degrees) A square is a quadrilateral in which all four sides are congruent and all four angles are congruent.
- 11. From General to Specific Quadrilateral trapezoid kite parallelogram rhombus rectangle square More specific
- 12. Perimeter and Area • The perimeter of a plane geometric figure is a measure of the distance around the figure. • The area of a plane geometric figure is the amount of surface in a region. perimeter area
- 13. Triangle h b a c Perimeter = a + b + c Area = bh 2 1 The height of a triangle is measured perpendicular to the base.
- 14. Rectangle and Square w l s Perimeter = 2w + 2l Perimeter = 4s Area = lw Area = s2
- 15. Parallelogram b a h Perimeter = 2a + 2b Area = hb Area of a parallelogram = area of rectangle with width = h and length = b
- 16. Trapezoid c d a b Perimeter = a + b + c + d Area = b a Parallelogram with base (a + b) and height = h with area = h(a + b) But the trapezoid is half the parallelgram h(a + b) 2 1 h
- 17. Ex: Name the polygon 3 2 1 4 5 6 hexagon 1 2 3 4 5 pentagon
- 18. Ex: What is the perimeter of a triangle with sides of lengths 1.5 cm, 3.4 cm, and 2.7 cm? 1.5 2.7 3.4 Perimeter = a + b + c = 1.5 + 2.7 + 3.4 = 7.6
- 19. Ex: The perimeter of a regular pentagon is 35 inches. What is the length of each side? Perimeter = 5s 35 = 5s s = 7 inches s Recall: a regular polygon is one with congruent sides.
- 20. Ex: A parallelogram has a based of length 3.4 cm. The height measures 5.2 cm. What is the area of the parallelogram? 3.4 5.2 Area = (base)(height) Area = (3.4)(5.2) = 17.86 cm2
- 21. Ex: The width of a rectangle is 12 ft. If the area is 312 ft2, what is the length of the rectangle? 12 312 Area = (Length)(width) L = 26 ft Let L = Length L 312 = (L)(12) Check: Area = (Length)(width) = (12)(26) = 312
- 22. Circle • A circle is a plane figure in which all points are equidistance from the center. • The radius, r, is a line segment from the center of the circle to any point on the circle. • The diameter, d, is the line segment across the circle through the center. d = 2r • The circumference, C, of a circle is the distance around the circle. C = 2pr • The area of a circle is A = pr2. r d
- 23. Find the Circumference • The circumference, C, of a circle is the distance around the circle. C = 2pr • C = 2pr • C = 2p(1.5) • C = 3p cm 1.5 cm
- 24. Find the Area of the Circle • The area of a circle is A = pr2 • d=2r • 8 = 2r • 4 = r • A = pr2 • A = p(4)2 • A = 16p sq. in. 8 in
- 25. Composite Geometric Figures • Composite Geometric Figures are made from two or more geometric figures. • Ex: +
- 26. • Ex: Composite Figure -
- 27. Ex: Find the perimeter of the following composite figure + = 8 15 Rectangle with width = 8 and length = 15 Half a circle with diameter = 8 radius = 4 Perimeter of composite figure = 38 + 4p. Perimeter of partial rectangle = 15 + 8 + 15 = 38 Circumference of half a circle = (1/2)(2p4) = 4p.
- 28. Ex: Find the perimeter of the following composite figure 28 60 42 12 ? = a ? = b 60 a 42 60 = a + 42 a = 18 28 b 12 28 = b + 12 b = 16 Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176
- 29. Ex: Find the area of the figure 3 3 8 8 Area of rectangle = (8)(3) = 24 3 8 Area of triangle = ½ (8)(3) = 12 Area of figure = area of the triangle + area of the square = 12 + 24 = 36. 3
- 30. Ex: Find the area of the figure 4 3.5 4 3.5 Area of rectangle = (4)(3.5) = 14 4 Diameter = 4 radius = 2 Area of circle = p22 = 4p Area of half the circle = ½ (4p) = 2p The area of the figure = area of rectangle – cut out area = 14 – 2p square units.
- 31. Ex: A walkway 2 m wide surrounds a rectangular plot of grass. The plot is 30 m long and 20 m wide. What is the area of the walkway? 20 30 2 What are the dimensions of the big rectangle (grass and walkway)? Width = 2 + 20 + 2 = 24 Length = 2 + 30 + 2 = 34 The small rectangle has area = (20)(30) = 600 m2. What are the dimensions of the small rectangle (grass)? Therefore, the big rectangle has area = (24)(34) = 816 m2. The area of the walkway is the difference between the big and small rectangles: 20 by 30 Area = 816 – 600 = 216 m2. 2
- 32. Find the area of the shaded region 10 10 10 r = 5 Area of each circle = p52 = 25p ¼ of the circle cuts into the square. But we have four ¼ 4(¼)(25p ) cuts into the area of the square. Area of square = 102 = 100 Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25p square units r = 5