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: Point (Straight) Line Plane (Surface) 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 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. AB A B A line segment is part of a line and has two endpoints. AB A B

An angle is formed by two rays with the same endpoint. side side vertex 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 name of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon

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.

From General to Specific Quadrilateral trapezoid kite More specific 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

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

Rectangle and Square Perimeter = 2w + 2l Perimeter = 4s Area = lw Area = s2

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

Trapezoid 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  hexagon 5 4 1 2 5  pentagon 3 4

Ex: What is the perimeter of a triangle with sides of lengths 1 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 Perimeter = a + b + c 3.4 = 1.5 + 2.7 + 3.4 = 7.6

Ex: The perimeter of a regular pentagon is 35 inches 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 = 5s 35 = 5s s = 7 inches

Ex: A parallelogram has a based of length 3. 4 cm 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 Area = (3.4)(5.2) = 17.86 cm2 3.4

Ex: The width of a rectangle is 12 ft Ex: The width of a rectangle is 12 ft. If the area is 312 ft2, what is the length of the rectangle? Area = (Length)(width) 312 12 Let L = Length L 312 = (L)(12) L = 26 ft Check: Area = (Length)(width) = (12)(26) = 312

Circle r 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 = 2r The circumference, C, of a circle is the distance around the circle. C = 2pr The area of a circle is A = pr2.

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

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

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 Half a circle with diameter = 8  radius = 4 Circumference of half a circle = (1/2)(2p4) = 4p. Perimeter of partial rectangle = 15 + 8 + 15 = 38 Perimeter of composite figure = 38 + 4p.

Ex: Find the perimeter of the following composite figure 60 12 28 42 ? = b 12 28 ? = a 60 b a 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 Area of figure = area of the triangle + area of the square = 12 + 24 = 36. 3 8 Area of rectangle = (8)(3) = 24

Ex: Find the area of the figure 4 4 3.5 3.5 Area of rectangle = (4)(3.5) = 14 The area of the figure = area of rectangle – cut out area = 14 – 2p square units. 4 Diameter = 4  radius = 2 Area of circle = p22 = 4p  Area of half the circle = ½ (4p) = 2p

Ex: A walkway 2 m wide surrounds a rectangular plot of grass 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? What are the dimensions of the big rectangle (grass and walkway)? 2 30 Width = 2 + 20 + 2 = 24 Length = 2 + 30 + 2 = 34 20 2 Therefore, the big rectangle has area = (24)(34) = 816 m2. What are the dimensions of the small rectangle (grass)? 20 by 30 The small rectangle has area = (20)(30) = 600 m2. The area of the walkway is the difference between the big and small rectangles: Area = 816 – 600 = 216 m2.

Find the area of the shaded region 10 10 Area of square = 102 = 100 Area of each circle = p52 = 25p r = 5 ¼ of the circle cuts into the square. But we have four ¼ 4(¼)(25p ) cuts into the area of the square. 10 r = 5 Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25p square units