INTRODUCTION TO GEOMETRY Geometry The word geometry comes

INTRODUCTION TO GEOMETRY

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

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.

The Elements • Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms: Ø Point is that which has no part Ø (Straight) Line is a line that lies evenly with the points on itself Ø Plane (Surface) is a plane that lies evenly with the straight lines on itself Actually, Euclid did attempt to define these basic terms. . .

Basic Terms & Definitions • A ray starts at a point (called the endpoint) and extends indefinitely in one direction. A B AB • A line segment is part of a line and has two endpoints. A B AB

• An angle is formed by two rays with the same endpoint. side vertex side • An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees.

• A right angle has a measure of 90 degrees. • A straight angle has a measure of 180 degrees.

• 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.

Polygons # of sides 3 4 5 6 7 8 9 10 name of Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon

Quadrilaterals • Recall: a quadrilateral is a 4 -sided polygon. We can further classify quadrilaterals: q A trapezoid is a quadrilateral with at least one pair of parallel sides. q A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. q A kite is a quadrilateral in which two pairs of adjacent sides are congruent. q A rhombus is a quadrilateral in which all sides are congruent. q A rectangle is a quadrilateral in which all angles are congruent (90 degrees) q A square is a quadrilateral in which all four sides are congruent and all four angles are congruent.

From General to Specific More specific Quadrilateral trapezoid kite parallelogram rhombus rectangle square

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. area perimeter

c a Triangle h b Perimeter = a + b + c Area = bh The height of a triangle is measured perpendicular to the base.

Rectangle and Square s w l Perimeter = 2 w + 2 l Perimeter = 4 s Area = lw Area = s 2

Parallelogram a h b Perimeter = 2 a + 2 b Area = hb Area of a parallelogram = area of rectangle with width = h and length = b

a c Trapezoid b d h b a Perimeter = a + b + c + d Area = h(a + b) Parallelogram with base (a + b) and height = h with area = h(a + b) But the trapezoid is half the parallelgram

Ex: Name the polygon 2 1 6 3 5 hexagon 4 1 2 5 pentagon 3 4

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

Ex: The perimeter of a regular pentagon is 35 inches. What is the length of each side? s Recall: a regular polygon is one with congruent sides. Perimeter = 5 s 35 = 5 s s = 7 inches

Ex: A parallelogram has a based of length 3. 4 cm. The height measures 5. 2 cm. What is the area of the parallelogram? Area = (base)(height) 5. 2 3. 4 Area = (3. 4)(5. 2) = 17. 86 cm 2

Ex: The width of a rectangle is 12 ft. If the area is 312 ft 2, what is the length of the rectangle? 12 312 Area = (Length)(width) Let L = Length L 312 = (L)(12) L = 26 ft Check: Area = (Length)(width) = (12)(26) = 312

r Circle d • 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 = 2 r • The circumference, C, of a circle is the distance around the circle. C = 2 pr • The area of a circle is A = pr 2.

Find the Circumference 1. 5 cm • The circumference, C, of a circle is the distance around the circle. C = 2 pr • C = 2 p(1. 5) • C = 3 p cm

Find the Area of the Circle • The area of a circle is A = pr 2 8 in • d=2 r • 8 = 2 r • 4=r • A = pr 2 • A = p(4)2 • A = 16 p sq. in.

Composite Geometric Figures • Composite Geometric Figures are made from two or more geometric figures. • Ex: +

• Ex: Composite Figure -

Ex: Find the perimeter of the following composite figure 15 8 = + Rectangle with width = 8 and length = 15 Perimeter of partial rectangle = 15 + 8 + 15 = 38 Half a circle with diameter = 8 radius = 4 Circumference of half a circle = (1/2)(2 p 4) = 4 p. Perimeter of composite figure = 38 + 4 p.

Ex: Find the perimeter of the following composite figure 60 12 28 42 ? =b 12 28 ? =a 60 a b 42 60 = a + 42 a = 18 28 = b + 12 b = 16 Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176

Ex: Find the area of the figure 3 3 3 8 Area of triangle = ½ (8)(3) = 12 8 3 8 Area of rectangle = (8)(3) = 24 Area of figure = area of the triangle + area of the square = 12 + 24 = 36.

Ex: Find the area of the figure 4 4 3. 5 Area of rectangle = (4)(3. 5) = 14 4 The area of the figure = area of rectangle – cut out area = 14 – 2 p square units. Diameter = 4 radius = 2 Area of circle = p 22 = 4 p Area of half the circle = ½ (4 p) = 2 p

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? 2 30 20 What are the dimensions of the big rectangle (grass and walkway)? Width = 2 + 20 + 2 = 24 Length = 2 + 30 + 2 = 34 2 Therefore, the big rectangle has area = (24)(34) = 816 m 2. What are the dimensions of the small rectangle (grass)? 20 by 30 The small rectangle has area = (20)(30) = 600 m 2. The area of the walkway is the difference between the big and small rectangles: Area = 816 – 600 = 216 m 2.

Find the area of the shaded region 10 10 Area of square = 102 = 100 10 Area of each circle = p 52 = 25 p r=5 ¼ of the circle cuts into the square. But we have four ¼ 4(¼)(25 p ) cuts into the area of the square. Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25 p square units