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 Geometry This section will familiarize you with the properties of angles, lines, polygons, triangles, and circles, as well as the formulas for area, volume, and perimeter. Geometry is the study of shapes and the relationships among them. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and con- cepts. Therefore, a list of definitions and important formulas is provided below. Geometry Terms Area the space inside a 2-dimensional figure Circumference the distance around a circle Chord a line segment that goes through a circle, with its endpoints on the circle Congruent lengths, measures of angles, or size of figures are equal Diameter a chord that goes directly through the center of a circle—the longest line segment that can be drawn in a circle Hypotenuse the longest side of a right triangle, always opposite the right angle Leg either of the two sides of a right triangle that make the right angle Perimeter the distance around a figure π (pi) The ratio of any circle’s circumference to its diameter. Pi is an irrational number, but most of the time it is okay to approximate π with 3.14. Radius a line segment from the center of a circle to a point on the circle (half of the diameter) Surface Area the sum of the areas of all of a 3-dimensional figure’s faces Vol u me the space inside a 3-dimensional figure – THEA MATH REVIEW– 140 Coordinate Geometry Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal (x) axis and a vertical (y) axis. These two axes intersect at one coordinate point, (0,0), the origin. A coordinate point, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and the sec- ond number representing the vertical placement. Coordinate points are given in the form of (x,y). Graphing Ordered Pairs (Points) ■ The x-coordinate is listed first in the ordered pair and tells you how many units to move to either the left or to the right. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left. ■ The y-coordinate is listed second and tells you how many units to move up or down. If the y-coordinate is positive, move up. If the y-coordinate is negative, move down. Example Graph the following points: (2,3), (3,–2), (–2,3), and (–3,–2). y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1–2–3–4 x 5 origin (0,0) – THEA MATH REVIEW– 141 ■ Notice that the graph is broken up into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs based on their signs: Lines, Line Segments, and Rays A line is a straight geometric object that goes on forever in both directions. It is infinite in length, and is repre- sented by a straight line with an arrow at both ends. Lines can be labeled with one letter (usually in italics) or with two capital letters near the arrows. Line segments are portions of lines. They have two endpoints and a definitive length. Line segments are named by their endpoints. Rays have an endpoint and continue straight in one direc- tion. Rays are named by their endpoint and one point on the ray. m A B A B A B Line m Line AB (AB) ↔ → Line segment AB (AB) Ray AB (AB) − Points Sign of Quadrant Coordinates (2,3) (+,+) I (–2,3) (–,+) II (–3,–2) (–,–) III (3,–2) (+,–) IV II I III IV (−2,3) (2,3) (−3,−2) (3,−2) – THEA MATH REVIEW– 142 PARALLEL AND PERPENDICULAR LINES Parallel lines (or line segments) are a pair of lines, that if extended, would never intersect or meet. The symbol || is used to denote that two lines are parallel. Perpendicular lines (or line segments) are lines that intersect to form right angles, and are denoted with the symbol ⊥. LENGTHS OF HORIZONTAL AND VERTICAL SEGMENTS Two points with the same y-coordinate lie on the same horizontal line and two points with the same x-coordi- nate lie on the same vertical line. The distance between a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points. Example Find the lengths of line segments AB ៮ and BC ៮ . | 2 – 7 | = 5 = AB ៮ | 1 – 5 | = 4 = BC ៮ (2,1) (7,5) C BA C D A B C D A B Parallel Lines AB and CD AB ʈ CD ↔↔ Parallel Lines AB and CD AB Ќ CD ↔↔ – THEA MATH REVIEW– 143 DISTANCE OF COORDINATE POINTS The distance between any two points is given by the formula d = ͙(x 2 – x ෆ 1 ) 2 + (y ෆ 2 – y 1 ) 2 ෆ ,where (x 1 ,y 1 ) repre- sents the coordinates of one point and (x 2 ,y 2 ) is the other. The subscripts are used to differentiate between the two different coordinate pairs. Example Find the distance between points A(–3,5) and B(1,–4). Let (x 1, y 1 ) represent point A and let (x 2 ,y 2 ) represent point B. This means that x 1 = –3, y 1 = 5, x 2 = 1, and y 2 = –4. Substituting these values into the formula gives us: d = ͙(x 2 – x ෆ 1 ) 2 + (y ෆ 2 – y 1 ) 2 ෆ d = ͙(–3 – 1 ෆ ) 2 + (5 ෆ – (–4) ෆ ) 2 ෆ d = ͙(–4) 2 + ෆ (9) 2 ෆ d = ͙16 + 8 ෆ 1 ෆ d = ͙97 ෆ MIDPOINT The midpoint of a line segment is a point located at an equal distance from each endpoint. This point is in the exact center of the line segment, and is said to be equidistant from the segment’s endpoints. In coordinate geometry, the formula for finding the coordinates of the midpoint of a line segment whose endpoints are (x 1 ,y 1 ) and (x 2 ,y 2 ) is given by M = ( ᎏ x 1 + 2 x 2 ᎏ , ᎏ y 1 + 2 y 2 ᎏ ). Example Determine the midpoint of the line segment AB ៮ with A(–3,5) and B(1,–4). Let (x 1 ,y 1 ) represent point A and let (x 2 ,y 2 ) represent point B. This means that x 1 = –3, y 1 = 5, x 2 = 1, and y 2 = –4. Substituting these values into the formula gives us: M A B M is the midpoint of AB − – THEA MATH REVIEW– 144 M = ( ᎏ –3 2 +1 ᎏ , ᎏ 5+ 2 (–4) ᎏ ) M = (– ᎏ 2 2 ᎏ , ᎏ 1 2 ᎏ ) M = (–1, ᎏ 1 2 ᎏ ) Note: There is no such thing as the midpoint of a line, as lines are infinite in length. SLOPE The slope of a line (or line segment) is a numerical value given to show how steep a line is. A line or segment can have one of four types of slope: ■ A line with a positive slope increases from the bottom left to the upper right on a graph. ■ A line with a negative slope decreases from the upper left to the bottom right on a graph. ■ A horizontal line is said to have a zero slope. ■ A vertical line is said to have no slope (undefined). ■ Parallel lines have equal slopes. ■ Perpendicular lines have slopes that are negative reciprocals of each other. Positive slope y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 3 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 2 –5 –1 –2 –3 –4 1 5432 –5 –1 –2 –3–4 x 5 y 1 4 3 2 –5 –1 –2 –3 –4 1 542 –5 –1 –2 –3–4 x 5 3 3 Negative slope Zero slope Undefined (no) slope – THEA MATH REVIEW– 145 . slope y 1 4 3 2 –5 1 –2 –3 –4 1 5432 –5 1 –2 –3–4 x 5 y 1 4 3 2 –5 1 –2 –3 –4 1 5432 –5 1 –2 –3–4 x 5 y 1 4 2 –5 1 –2 –3 –4 1 5432 –5 1 –2 –3–4 x 5 y 1 4 3 2 –5 1 –2 –3 –4 1 542 –5 1 –2 –3–4 x 5 3 3 Negative. whose endpoints are (x 1 ,y 1 ) and (x 2 ,y 2 ) is given by M = ( ᎏ x 1 + 2 x 2 ᎏ , ᎏ y 1 + 2 y 2 ᎏ ). Example Determine the midpoint of the line segment AB ៮ with A(–3,5) and B (1, –4). Let (x 1 ,y 1 ) represent. following points: (2,3), (3,–2), (–2,3), and (–3,–2). y 1 4 3 2 –5 1 –2 –3 –4 1 5432 –5 1 2–3–4 x 5 origin (0,0) – THEA MATH REVIEW– 14 1 ■ Notice that the graph is broken up into four quadrants