Geometry A & Geometry B & Honors Geometry Summer Packet

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Phillip Andrews
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1 Geometry A & Geometry B & Honors Geometry Summer Packet RADICALS Radical expressions contain numbers and/or variables under a radical sign,. A radical sign tells you to take the square root of the value under the symbol. The radicand is the expression under the radical sign. EXAMPLE 1. SIMPLIFY a) 18 b) ) 7 ) 80 ) 80 4) 00 Product Property of Square Roots: For any numbers a and b where a 0 and b 0, ab a b. EXAMPLE. SIMPLIFY a) 7 7 1) 7 49 ) Quotient Property of Square Roots: For any numbers a and b where a a a 0 and b> 0,. b b Pg. 1

2 EXAMPLE. SIMPLIFY a) b) ) 0 ) 4 16 Rationalizing the Denominator: This method may be used to remove or eliminate radicals from the denominator of a fraction. EXAMPLE 4. SIMPLIFY a) b) ) 7 ) ) 11 Pg.

3 THE PYTHAGOREAN THEOREM If a and b are the measure of the legs of a right triangle and c is the measure of the hypotenuse, then a + b c. a c You use the Pythagorean Theorem to find the length of the missing side of a right triangle. b EXAMPLE 1. FIND THE LENGTH OF THE MISSING SIDE OF A RIGHT TRIANGLE IF a1 AND b. 1 + c c 169 c 169 c ± 1 c Yet, you only keep +1 because c is a distance measure. 1) a4 and b ) b8 and c10 ) a7 and b4 Pg.

4 DISTANCE FORMULA This formula is used to find the distance d between any two points with coordinates (x 1, y 1 ) 1 1. and (x, y ). It is found by the following formula: d ( x x ) + ( y y ) EXAMPLE 1. FIND THE DISTANCE BETWEEN THE POINTS WITH COORDINATES (, ) AND (6, 4). x, y, and x, y 6, 4 let ( ) ( ) ( ) ( ) 1 1 ( 6 ) ( 4 ) d + ( ) ( 1) ) Find the distance between the points with coordinates (-, 4) and (-1, -). ) Find the distance between the points with coordinates (-, 4) and (, ). Pg. 4

5 MIDPOINT FORMULA This formula is used to find the midpoint M between any two points with coordinates x1+ x y1+ y (x 1, y 1 ) and (x, y ). It is found by the following formula: M,. EXAMPLE 1. FIND THE MIDPOINT BETWEEN THE POINTS WITH COORDINATES (4, ) AND (6, 4). x, y 4, and x, y 6, 4 let ( ) ( ) ( ) ( ) M, 9, 1) Find the midpoint between the points with coordinates (, 4) and (-, ). ) Find the midpoint between the points with coordinates (-, 4) and (-1, -). Pg.

6 THE COORDINATE PLANE When we graph ordered pairs (points), we have an x and a y value to graph. The ordered pair is represented by ( x, y ). We graph this coordinate on a Coordinate Plane. The x represents the right or left movement, which is counted first. Then, the y represents the up or down movement, which is counted second. The point will end up in one of the four Quadrants which are sections of the Coordinate Plane. Usually, these Quadrants are labeled with Roman Numerals. EXAMPLE 1. PLOT THE POINTS AND IDENTIFY THE QUADRANT OR AXIS. a) (,) is in Quadrant I b) (,) is in Quadrant II c) ( 4, 1) is in Quadrant III d) (, 1) is in Quadrant IV e) (0,) is on the y-axis f) (,0) is on the x-axis Pg. 6

7 1) (, ) is in Quadrant ) (0,) is on the axis ) (4,1) is in Quadrant 4) (, ) is in Quadrant ) (,0) is on the axis 6) (, ) is in Quadrant Pg. 7

8 SLOPE The ratio, m, of the rise to the run as you move from one point to another along a line. rise m run rise: vertical change run: horizontal change rise run Determining Slope Given Two Points: Given the coordinates of two points (, ) y y1 x y on a line the slope can be found as follows: m. x x (, ) 1 x y and EXAMPLE 1. DETERMINE THE SLOPE OF THE LINE THAT PASSES THROUGH (, -) AND (7, -10). let ( x1, y1) (, ) and ( x, y ) ( 7, 10) m 10 ( ) 7 1 1) Find the slope of the line that passes through (4, 6) and (10, -). ) Find the slope of the line that passes through (, -1) and (, -). 1 1 Pg. 8

9 Point-Slope Form: For a given point ( 1, 1) form is as follows y y m ( x x ) 1 1. LINEAR EQUATIONS x y on a nonvertical line having slope m, the point-slope EXAMPLE 1. WRTIE THE POINT-SLOPE FORM OF AN EQUATION FOR A LINE THAT PASSES THROUGH (-, ) AND HAS A SLOPE OF. 4 let ( x1, y1) (, ) and m 4 y ( x+ ) 4 1) Write the point-slope form of an equation for a line that passes through (, 8) and has a slope of. slope of. ) Write the point-slope form of an equation for a line that passes through (-6, 1) and has a Standard Form: The standard form of a linear equation is Ax+ByC, where A, B, and C are integers, A 0, and A and B are not both zero. + IN STANDARD FORM. 4 4( y+ ) 4 ( x ) 4 4y+ 0 ( x ) 4y+ 0 x+ 10 4y x y x 10 x+ 4y 10 EXAMPLE. WRITE y ( x ) + in standard form. 4 1) Write y ( x 1) ) Write y ( x 9) in standard form. Pg. 9

10 Slope-Intercept Form: Given the slope m and the y-intercept b of a line, the slope-intercept form of an equation of the line is y mx + b. EXAMPLE. WRITE AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM IF THE LINE HAS A SLOPE OF AND A Y-INTERCEPT OF 6. let m and b 6 y x + 6 1) Write an equation of a line in slope-intercept form if the line has a slope of and a y-intercept of. ) Write an equation of a line in slope-intercept form if the line has a slope of 1 4 and a y-intercept of -10. EXAMPLE 4. FIND THE SLOPE AND Y-INTERCEPT OF THE GRAPH OF x y 6. Turn the equation into slope-intercept form. y x+ 6 y x 6 + y x m and b 1) Find the slope and y-intercept of the graph of x+ 4y 10. ) Find the slope and y-intercept of the graph of 1 4x y. Pg. 10

11 ABSOLUTE VALUE An absolute value of a number is the distance the number is from zero. Therefore, the solution of an absolute value is always positive due to the fact that it represents a distance. The symbol for absolute value is. EXAMPLE 1. FIND THE ABSOLUTE VALUE. a) b) c) d) ) 7 ) 7 ) 1 8 4) 11 Pg. 11

12 SOLVING EQUATIONS EXAMPLE 1. SOLVE EACH EQUATION FOR THE VARIABLE. a) m + 6 b) x 8 m 6 x 8+ m 4 x 6 x x 6 1) + g 7 ) 9+ s ) a ( 6) 4) d + ( 44) 61 EXAMPLE. SOLVE EACH EQUATION FOR THE VARIABLE. c a) p b) 7 9 p c ( 9) p 7 c 41 1) x 4 ) 6a f ) 6 4) x 97 ( ) EXAMPLE. SOLVE EACH EQUATION FOR THE VARIABLE. a) 6x + 7 8x 1 b) 8(4 + 9 x) 7( 11 x) x 6x 7x 14 77x 0 x 77x 7x 14 0 x x 18 x x 18 x 1) 17 + x 1+ x ) 7 x x 4( + x) 1 x ) 7(x ) ( 1x + 0) 4) + 6 ( x ) Pg. 1

13 SOLVING SYSTEMS OF EQUATIONS Sometimes when solving equations there are two or more variables we need to solve for. Most of the time in Geometry, you will be given two equations with two variables which is known as a system of equations. You are looking for the coordinate, ( x, y), that works for both equations. In Algebra, you should have learned three different methods for solving systems. The easiest methods are to use Substitution or Addition/Elimination. EXAMPLE 1. SOLVE THE SYSTEM USING SUBSTITUTION. x + y a) 4x + 4 y 8 First, you need to solve for one of the variables, that means get one variable by itself. Sometimes this step is already done for you. The obvious choice would be to either solve for x or y in the first equation. Let s solve for y in the first equation. y x Now, go the second equation and substitute x in place of the variable y. 4x + 4( x) 8 Then, solve for x. 4x + 1 8x 8 4x x 8 1 4x 4 4x x 1 Finally, substitute 1 in place of the variable x in either of the two original equations. It is usually easiest to substitute the value into the equation that you simplified. y (1) y y 1 Thus, your answer is ( 1,1 ). Pg. 1

14 EXAMPLE. SOLVE THE SYSTEM USING ADDITION/ELIMINATION. x + y a) 4x + 4 y 8 First, you need to add both equations together, so that either the x or the y cancels out. If a variable won t cancel when adding the original equations, you need to change the equations. You do this by multiply one or both of the equations by numbers. In this case, you can either turn the x in the top equation into -4x, or you can turn the y in the top equation into -4y. Let s cancel the x values. Thus, you need to multiply the entire top equation by -. ( x + y )( ) 4x y 6 Now, add the two equations together. The x will cancel out in both equation. 4x y 6 4x + 4 y 8 0x + y y Then, solve for the remaining variable, y. y y 1 Once you have solved for one variable, substitute the value of the variable into either of the two original equations. Let s go back to the first equation and substitute 1 for y. x + 1 Finally, solve for x. x 1 x x x 1 Thus, your answer is ( 1,1 ).. SOLVE EACH SYSTEM OF EQUATIONS USING EITHER METHOD. y x 4 x y 8 1) ) x + y x + y 9 Pg. 14

15 Geometry Summer Packet Answers RADICALS EXAMPLE 1 1) ) 4 ) 70 4) 10 EXAMPLE 1) 1 7 ) 4 EXAMPLE 1) ) 1 EXAMPLE 4 1 1) 7 ) 6 ) 8 THE PYTHAGOREAN THEOREM EXAMPLE 1 1) c ) a 6 ) c DISTANCE FORMULA EXAMPLE 1 1) 8 ) 17 EXAMPLE 1 1) (-1, ) ) MIDPOINT FORMULA 1, THE COORDINATE PLANE EXAMPLE 1 1) III ) y ) I 4) II ) x 6) IV Pg. 1

16 EXAMPLE 1 1) m ) m SLOPE EXAMPLE 1 1) y 8 ( x ) ) LINEAR EQUATIONS y 1 ( x + 6) EXAMPLE 1) x + 4 y 9 ) x + y 1 EXAMPLE 1) y x + ) 1 y x 10 4 EXAMPLE 4 1) m and b ) m 1 and b 6 4 ABSOLUTE VALUE EXAMPLE 1 1) 7 ) 7 ) 4 4) 1 SOLVING EQUATIONS EXAMPLE 1 1) y 9 ) s 14 ) a 11 4) d 17 Pg. 16

17 EXAMPLE 1) x 8 ) EXAMPLE 1) x 4 ) x 1 ) EXAMPLE 1, 7 1) ( ) ) (, ) 1 a ) f 88 4) x 8148 x 4) SOLVING SYSTEMS OF EQUATIONS 9 x 1 Pg. 17

Summer Packet Pre-AP Algebra

Name: (5/11/17) Summer Packet Pre-AP Algebra 1-2018-19 To receive credit all work must be shown. There will be a test the first week back from summer on this packet. Any work done on additional paper must