Geometry Honors Summer Packet Hello Student, First off, welcome to Geometry Honors! In the fall, we will embark on an eciting mission together to eplore the area (no pun intended) of geometry. This packet is full of stimulating math problems for you to enjoy over the summer. The purpose of this summer packet is to review the algebra and geometry topics you have already mastered in mathematics and to make sure you are prepared for Geometry Honors. An honors level course means that we will move at a fast pace and you will be epected to use your prior knowledge and apply it to the geometric world net school year. This packet contains many sections of material you should have seen before. You must complete the practice problems in each section and be prepared to turn them into your teacher. This packet will be due during the first week of school and you will receive a grade for completing it. There is not a test or quiz specifically on this material but these skills will be used throughout the year not only on problems in class but on assessments as well. If you need help while working on this packet, you can visit a couple of math websites such as the following: http://www.purplemath.com/modules/inde.htm or another useful site is https://www.khanacademy.org/. Time will not be spent in class going over these topics so when school begins you can use the Titan Learning Center (TLC) in the library to obtain additional help. You can also see your teacher for assistance. These problems should be done without a calculator. The answers are provided at the back of the packet but in math, the work is usually more important than the answer. Show enough work appropriate to each problem to receive full credit and bo your final answer. Do not wait until the last minute to complete these, as August will be here before you know it! You should pace yourself and consider completing a little each week but it is up to you to organize your time. Try and keep track of how much time it takes and what problems you had any difficulties with so you can answer the questions at the very end. Etra copies of the packets will be available on the school website at www.glenbrook5.org. We look forward to seeing you in August ready and ecited to learn. Make sure you bring you packet on the first full day of school! Looking forward to meeting you, ~ Carrie Fraher, Marianne Kerr, & John O Malley Glenbrook South Geometry Honors Teachers
1. Solve the following for the variable: You should know how to solve equations with fractions as well as proportions. a. p + 5 = 1 b. u+ ( u ) = 10 c. 180 = (90 ) d. 1 1 ( 7) = 8 e. + = f. q = 4 5 0 g. y + 7 6 H = h. 1 = i. 6 + 10 = 70 y j. = 64 k. ( 4) = 15 l. 100π = πr
. Use the distributive property to multiply the following epressions: (Write your answer in descending degree this is called the standard form for polynomials.) You need to be able to simplify epressions, especially binomials. a. ( )( 7) + b. y( y 5y 10) c. ( 4)( 4) + + d. ( 4) + (Hint: NOT + 16) e. ( ) f. ( )( 4 + 5). Factor the following polynomials: You need to be able to factor all different forms eplicitly focusing on quadratics. a. 9 b. 5 8 c. + 7+ 1 d. + 7 60 e. 10+ 5 f. 81 g. h. 49 9 + i. 4n 15n 5
4. Solve the following by factoring and the zero product property: You need to be able to solve equations by factoring a. ( )( ) 8 + 5 = 0 b. + 14+ 1 = 0 c. + 7= 8 Hint: Not In Standard Form d. 5 1 + + = e. 14 40 9 + = f. + + = 6 7 0 g. 11 10 0 + = h. 0 60 5 = + + i. 0 + = 5. Simplify the following fractions (you may have to think about factoring ) a. 5 10 15 + 6 b. 6 c. 6 4 + 8 6. Solve the following systems algebraically and show your work. You need to be able to solve systems using substitution and elimination methods. a. y = + 11 y = + 5 + y = 17 b. y = 5
+ 4y = c. = y + 7 d. + 5y = 6 y = 5 e. 4( y) = 0 ( 5 y) = 0 f. y = 5 1 y 1= 6 7. Graph the following on the grids provided by using at least two points and a straight line: You need to be able to graph lines in various forms. a. y = 5 b. 8y = 4 c. y 4 = ( + 4) (This is slope-intercept form) (This is standard form) (This is point-slope form)
d. = 5 e. y = 7 f. y 9 = 4( ) 8. Find the solution to the following systems by graphing: You need to be able to solve systems by graphing in addition to the algebraic methods you already practiced y = 4+ 5 = 4 a. 5 b. y = 6 y = 8 y = Solution: (, ) Solution: (, ) c. y 4= = 4y Solution: (, )
9-11. Write equations of lines through the specified form. You need to be able to write equations of lines given various information. 9. Write an equation of a line in slope-intercept form y = m + b for the following: a. The slope of the line is - and the y-intercept is 5. b. The line goes through the points (, 1) and (0, 7). Note: Question 15 has a formula you may need c. The lines that is parallel to y = 5 4 and goes through the point (7, -). 10. Write an equation of a line in point-slope form y y1 = m ( 1) for the following: a. The slope of the line is 7 and it goes through the point (, ). b. The line goes through the points (, -7) and (-4, 9). c. The line that is perpendicular to 1 y = + 8 and goes through the point (-8, -). 11. Use any form you want for the following: a. Write an equation of a vertical line through the point (4, -5). b. Write an equation of a horizontal line through the point (4, -5). c. Write an equation of a line with an undefined slope through the point (4, -5). d. Write an equation of a line with a zero slope through the point (4, -5). e. Write an equation of a line perpendicular to 6 4y = 1 and through the point (7, -5).
1. Simplify the following radicals: You should know how to perform operations (adding, subtracting, multiplying, & dividing) with radicals. a. 5 b. 4 c. 108 d. 49 e. 5 18 f. 50 g. 7 + 4 75 h. 9 0 45 i. 7 4 48 + 54 j. 4 k. 8 6 l. m. 4 5 5 n. 8 4 8 o. 7 7 p. ( ) 7 q. ( 5) r. 15 Note: You rationalize the denominator here s. 14 t. 19 u. 7 10
1. Evaluate each of the epressions below: a. ab when a = and b = 4 b. π rl when 11 r = and l = 6 c. y when = and y = 4 d. 1 π when r = 6 and h = 4 rh 14. Find the distance between the numbers on the number lines below. a. b. 1-7 9 c. The distance between two points on a number line is 8. One of the numbers is at -. Find all possibilities for the location of the other number. (Hint: Draw a picture) 15. For the following pairs of points find the following: a. The distance between them b. The coordinate of the midpoint between them c. The slope of the line on which they lie Remember that for any points ( 1, y 1) and (, y ) the following formulas are true: + y + y y y y Distance = ( 1) + ( y y1) Coordinate of Midpoint =, Slope = = Note: These are equations that you should have memorized and know how to use Pair 1: (, 4) and (15, 0) 1 1 1 1 Pair : (-, 6) and (7, 0)
16. In geometry, you might have learned that two angles are considered complimentary if they have a sum of 90. Two angles are considered supplementary if they have a sum of 180. a. If m 1 = 7 and m = then determine the value of if the angles are complimentary. b. If m 1 = 7 and m = then determine the value of if the angles are supplementary. c. If m A= + 16 and m B= 16 then find the numerical m B if the angles are supplementary. d. Find the measure of an angle that is twice as large as its supplement. (Hint: Try writing an equation) 17. In geometry, you might have also learned that angles with certain measures have specific names. - Acute angles are angles that have a measure between 0 and 90 or for an acute angle : 0 < < 90 - Right angles are angles that have a measure eactly 90 or for right angle : m = 90 o These are usually indicated with a bo to show they are right angles. o A perpendicular sign that looks like this: (upside down T) and can also indicate right angles - Obtuse angles are angles that have a measure between 90 and 180 or for obtuse angle : 90 < < 180 - Straight angles are angles that have a measure eactly 180 or for straight angle : m = 180 o In a diagram this would appear to be a line. - Vertical angles are non-adjacent angles formed by the intersection of two lines. o Angles that are vertical angles are opposite or across from each other and have the same measure. a. Find the value of below b. Find the value of below c. The angle below is obtuse. Give a range of possible values 5 + 0 5 + y 15y 5
18. The Pythagorean Theorem (possibly the coolest of all geometry theorems we will discuss) states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. In algebraic form where the hypotenuse of a right triangle is c and the legs are a and b this looks like: c = a + b a. If the legs of a right triangle are 5 and 1, then b. If the legs of a right triangle are and 7, then set up an equation and solve for the hypotenuse. set up an equation and solve for the hypotenuse. c. If the hypotenuse of a right triangle is 10 and a leg is 5, then find the value of the missing leg. Draw a picture as well with proper labels. d. Find (Hint: Look back at 4 in the packet) e. Find (Additional Note: Remember that side lengths cannot be negative)
19. You have learned about areas of shapes in your previous math eperiences. Here are the formulas again in case you forgot some of them. When we study this unit, you will be epected to memorize these. Area of a square: A= s where s is a side length bh Area of a triangle: A = where b is a base and h is a height Area of a circle: A= π r where r is the radius of the circle Area of a rectangle: A = bh where b is a base and h is a height Area of a trapezoid: hb ( 1+ b) A = where h is a height, b1 is a base and b is the other base Circumference of a circle C = π d or C = π r where d is the diameter of the and r is the radius of the circle Area of a parallelogram: A = bh where b is a base and h is a height Area of a kite/rhombus: dd 1 A = where d1 is a diagonal length and d is the other diagonal For circles: The diameter of a circle is twice the length of the radius a. Find the area of a square with a side of 4 b. Find the area of a triangle with a base of 10 and height of 4 c. A parallelogram has a base of 6 inches and an area of 54 in. Find the height. Sketch the diagram and indicate which segments are the base and height. d. A trapezoid has one base of 4 cm, a height of 8 cm, and an area of 6 cm. Find the length of the other base. e. The shape below is an isosceles trapezoid since the two legs are congruent. Find the area of the trapezoid. (Hint: Try drawing in an etra segment or two and using a theorem that is pretty cool) 10 15 15 8
f. Find the eact area of a circle that has a circumference of 4π. Note: Eact means in terms of pi. g. Find the area of the kite below. Pay attention to the special markings: sides with the same markings mean that they have the same length. 8 17 10 0. Please answer the following honestly to assist your teacher in making sure you are prepared: a. Overall, how did you feel about the material in this packet? b. Were there any topics that appeared new to you? If so, which ones? c. Which topics, if any, did you struggle with? d. How long did it take you to complete this packet? Give the best estimate you can. e. Any other comments?
Answers 1a. p = 4 1b. u = 1c. = 45 1d. = 9 1e. 1g. y = 1 1h. H = 4 1i. = 8 1j. 8 9 = 14 1f. q = 1 = ± 1k. = ± 1l. r = ± 10 a. e. 4 1 b. y 5y 10y c. 4 1 9 + f. + + 4 8 1 10 15 + 8+ 16 d. + 8+ 16 a. (9 1) b. 8 ( + )( ) c. ( + )( + 4) d. ( + 1)( 5) e. ( 5) f. ( + 9)( 9) g. ( + 7)( 7) h. ( )( + ) i. (4n+ 5)( n 5) 5 4a., 4b. -1, -1 4c. -8, 1 4d. -4, - 4e. 7 8 4f. -9, -4 4g. 5, 4h. 5, 1 4i. 0, 1, 1 5a. 5b. 1 6 5c. ( 4) 4 6a. (-, 7) 6b. 11, 6c. (5, -) 6d. (7, 8) 6e. (0, 0) 6f. 1, 7a. 7b. 7c. 7d. 7e. 7f.
8a. (, -) 8b. (-4, 6) 8c. (-8, -) 9a. y = + 5 9b. y = + 7 9c. y = 5 8 10a. y = 7( ) 10b. 8 y 9 = ( + 4) OR 8 y+ 7 = ( ) 10c. y+ = ( + 8) 11a. = 4 11b. y = 5 11c. = 4 11d. y = 5 11e. y+ 5 = ( 7) 1a. 5 1b. 6 1c. 6 1d. 1 1e. 15 1f. 10 1g. 1h. 9 5 1i. 6 4 1j. 1k. 4 1l. 6 1m. 40 1n. 64 6 1o. 147 1p. 147 1q. 45 1r. 5 1s. 7 1t. 19 1u. 7 5 1a. 4 1b. π 1c. -4 1d. π 14a. 9 14b. 16 14c. -11 or 5 15. Pair 1: a. 0 b. (9, 1) c. 4 Pair : a. 1 b. 5, c. 16a. 18 16b. 108 16c. 104 16d. 10 17a. 0 17b. 7 17c. 95 185 < < 18a. 1 18b. 58 18c. 5 18d. 6 18e. 8 19a. 16 19b. 0 19. 9 19d. 5 19e. 8 19f. 144π 19g. 168