Mathematics exam 3 doc

the sum of its interior angles will equal 180 + 180 = 360 degrees. m∠a + m∠b + m∠c + m∠d = 360° Interior Angles To find the sum of the interior angles of any polygon, use this formula: S = 180(x − 2)°, with x being the number of polygon sides Example Find the sum of the angles in this polygon: S = (5 − 2) × 180° S = 3 × 180° S = 540° Exterior Angles Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees. Similar Polygons If two polygons are similar, their corresponding angles are equal and the ratios of the corresponding sides are in proportion. Example These two polygons are similar because their angles are equal and the ratios of the correspon- ding sides are in proportion. Parallelograms A parallelogram is a quadrilateral with two pairs of par- allel sides. In the figure above, line AB || CD and BC || AD. A parallelogram has: ■ opposite sides that are equal (AB = CD and BC = AD) ■ opposite angles that are equal (m∠a = m∠c and m∠b = m∠d) ■ consecutive angles that are supplementary (m∠a + m∠b = 180°, m∠b + m∠c = 180°, m∠c + m∠d = 180°, m∠d + m∠a = 180°) SPECIAL TYPES OF PARALLELOGRAMS ■ A rectangle is a parallelogram that has four right angles. D A B C AB = CD D A B C 60° 10 4 6 18 120° 60° 120° 5 2 3 9 b c d e a b d a c – MEASUREMENT AND GEOMETRY– 400 ■ A rhombus is a parallelogram that has four equal sides. ■ A square is a parallelogram in which all angles are equal to 90 degrees and all sides are equal to each other. D IAGONALS In all parallelograms, diagonals cut each other into two equal halves. ■ In a rectangle, diagonals are the same length. ■ In a rhombus, diagonals intersect to form 90-degree angles. ■ In a square, diagonals have the same length and intersect at 90-degree angles.  Solid Figures, Perimeter, and Area The GED provides you with several geometrical formu- las. These formulas will be listed and explained in this section. It is important that you be able to recognize the figures by their names and to understand when to use which formulas. Don’t worry. You do not have to mem- orize these formulas. They will be provided for you on the exam. To begin, it is necessary to explain five kinds of measurement: 1. Perimeter The perimeter of an object is simply the sum of all of its sides. 2. Area Area is the space inside of the lines defining the shape. 3. Volume Volume is a measurement of a three- dimensional object such as a cube or a = Area 6 7 4 10 Perimeter = 6 + 7 + 4 + 10 = 27 B C A D AC = DB and AC DB BC A D BD AC DC A B AC = DB D CB A AB = BC = CD = DA m∠A = m∠B = m∠C = m∠D D C B A AB = BC = CD = DA – MEASUREMENT AND GEOMETRY– 401 rectangular solid. An easy way to envision vol- ume is to think about filling an object with water. The volume measures how much water can fit inside. 4. Surface Area The surface area of an object measures the area of each of its faces. The total surface area of a rectangular solid is double the sum of the areas of the three faces. For a cube, simply multiply the surface area of one of its sides by six. 5. Circumference Circumference is the measure of the distance around the outside of a circle.  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 a hori- zontal (x) axis and a vertical (y) axis. These two axes inter- sect 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, or coordinate, representing the horizontal placement and the second number, or coordinate, representing the vertical place- ment. Coordinate points are given in the form of (x,y). Graphing Ordered Pairs The x-coordinate ■ The x-coordinate is listed first in the ordered pair and it 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 ■ 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). Notice that the graph is broken up into four quadrants with one point plotted in each one. II I III IV (−2,3) (2,3) (−3,−2) (3,−2) Circumference 4 4 Surface area of front side = 16 Therefore, the surface area of the cube = 16 x 6 = 96. – MEASUREMENT AND GEOMETRY– 402 This chart indicates which quadrants contain which ordered pairs based on their signs: 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-coordinate lie on the same vertical line. The length of a horizontal or vertical segment can be found by taking the absolute value of the difference of the two points or by counting the spaces on the graph between them. Example Find the lengths of AB ៮ ៮ ៮ and line BC ៮ ៮ ៮ . Solution: | 2 − 7 | = 5 = AB ៮ ៮ ៮ | 1 − 5 | = 4 = BC ៮ ៮ ៮ Midpoint To find the midpoint of a segment, use the following for- mula: Midpoint x = ᎏ x 1 + 2 x 2 ᎏ Midpoint y = ᎏ y 1 + 2 y 2 ᎏ Example Find the midpoint of line segment AB ៮ ៮ ៮ . Solution: Midpoint x = ᎏ 1+ 2 5 ᎏ = ᎏ 6 2 ᎏ = 3 Midpoint y = 2 + ᎏ 1 2 0 ᎏ = ᎏ 1 2 2 ᎏ = 6 Therefore the midpoint of AB ៮ ៮ ៮ is (3,6). (5,10) Midpoint (1,2) B A (2,1) (7,5) C BA Sign of Points Coordinates Quadrant (2,3) (+,+) I (–2,3) (–,+) II (–3,–2) (–,–) III (3,–2) (+,–) IV – MEASUREMENT AND GEOMETRY– 403 Slope The slope of a line measures its steepness. It is found by writing the change in the y-coordinates of any two points on the line, over the change of the corresponding x-coordinates. (This is also known as the rise over the run.) The last step is to simplify the fraction that results. Example Find the slope of a line containing the points (3,2) and (8,9). Solution: ᎏ 9 8 − − 2 3 ᎏ = ᎏ 7 5 ᎏ Therefore, the slope of the line is ᎏ 7 5 ᎏ . Note: If you know the slope and at least one point on a line, you can find the coordinates of other points on the line. Simply move the required units determined by the slope. In the last example, from (8,9), given the slope ᎏ 7 5 ᎏ , move up seven units and to the right five units. Another point on the line, thus, is (13,16). IMPORTANT INFORMATION ABOUT SLOPE ■ A line that rises from left to right has a positive slope and a line that falls from left to right has a negative slope. ■ A horizontal line has a slope of 0, and a vertical line does not have a slope at all—it is undefined. ■ Parallel lines have equal slopes. ■ Perpendicular lines have slopes that are negative reciprocals. (3,2) (8,9) – MEASUREMENT AND GEOMETRY– 404 B ASIC PROBLEM SOLVING in mathematics is rooted in whole number math facts, mainly addition facts and multiplication tables. If you are unsure of any of these facts, now is the time to review. Make sure to memorize any parts of this review that you find troublesome. Your ability to work with numbers depends on how quickly and accurately you can do simple mathematical computations.  Operations Addition and Subtraction Addition is used when you need to combine amounts. The answer in an addition problem is called the sum or the total. It is helpful to stack the numbers in a column when adding. Be sure to line up the place-value columns and to work from right to left. CHAPTER Number Operations and Number Sense A GOOD grasp of the building blocks of math will be essential for your success on the GED Mathematics Test. This chapter covers the basics of mathematical operations and their sequence, variables, inte- gers, fractions, decimals, and square and cube roots. 42 405 . down. Example Graph the following points: (2 ,3) , (3, −2), (−2 ,3) , and ( 3, −2). Notice that the graph is broken up into four quadrants with one point plotted in each one. II I III IV (−2 ,3) (2 ,3) ( 3, −2) (3, −2) Circumference 4 4 Surface. (3, 6). (5,10) Midpoint (1,2) B A (2,1) (7,5) C BA Sign of Points Coordinates Quadrant (2 ,3) (+,+) I (–2 ,3) (–,+) II ( 3, –2) (–,–) III (3, –2) (+,–) IV – MEASUREMENT AND GEOMETRY– 4 03 Slope The slope of a line measures its steepness ᎏ y 1 + 2 y 2 ᎏ Example Find the midpoint of line segment AB ៮ ៮ ៮ . Solution: Midpoint x = ᎏ 1+ 2 5 ᎏ = ᎏ 6 2 ᎏ = 3 Midpoint y = 2 + ᎏ 1 2 0 ᎏ = ᎏ 1 2 2 ᎏ = 6 Therefore the midpoint of AB ៮ ៮ ៮ is (3, 6). (5,10) Midpoint (1,2) B A (2,1) (7,5) C BA