Rising Geometry Students! Answer Key As a 7 th grader entering in to Geometry next year, it is very important that you have mastered the topics listed below. The majority of the topics were taught in Algebra 1, but it is important to remember some of the basic geometry you learned in elementary school. Geometric formulas for circles, rectangles and triangles (perimeter and area) Solving proportions Solving multistep equations and inequalities Equations of lines (slope, parallel lines, perpendicular lines) *Systems of equations *Factoring binomials and trinomials *Solving quadratics Simplifying radicals and rationalizing the denominator A * indicates the student must know how to factor and solve a system without the use of the calculator. Geometry is a highly accelerated, fast-paced high school level course. In order to cover the required curriculum, the Geometry teachers will not have time to reteach Algebra 1 content. We recommend completing this packet, a few questions at a time, over the course of the summer. Your Geometry teacher prefers that you do not complete the problems right after school ends. Everything is still fresh in your mind! Take a few weeks off, start up the packet at the beginning or middle of July, and see what you remember! We will be posting answer keys to each section throughout the summer. We will also be providing you with tutorial videos and extra practice. Please use the resources we have given you as the opportunity to review the material and challenge yourself along the way! If you have any questions, please do not hesitate to reach out to one of the Geometry teachers listed below. We look forward to meeting you! Jayme.Lorenz@lcps.org Christina.ODonovan@lcps.org Stephen.Obrien@lcps.org (Directions for accessing the website can be found on the next page.)
1. 1. Go to the staff directory on the Mercer webpage and find Lorenz, J 2. On the left-hand side choose Uprising 8 th Graders (make sure it drops down Geometry and Algebra 2/Trig tabs) 3. Select the Geometry tab. When you select the class you will see options for the packet, answer keys, etc. 4. Check back often so you don t miss anything!
Geometry Basics Write the following formulas in the boxes provided. Area of a circle: Circumference (perimeter) of a circle: Area of a rectangle: Perimeter of a rectangle: Area of a triangle: 1. Find the area of the circle. 2. Find the circumference of the circle. A = 38.47 units 2 C = 59.69 units 3. Find the area of the rectangle. 4. Find the perimeter of the rectangle. A = 90 units 2 P = 26 units 5. Find the area of the triangle. 6. Find the perimeter of the triangle. A = 85 cm 2 P = 136 km 7. Find the area of the shaded region. 8. Find the perimeter of the figure. A = 49.74 in 2 P = 64 units 9.A triangular sail has a perimeter of 25 meters. Side a is 2 meters shorter than twice side b, and side c is 3 meters longer than side b. Find the length of each side. a = 10 m b = 6 m c = 9 m
Solving Equations, Inequalities, and Proportions How do you solve a proportion? To solve a proportion, you cross multiply!(make sure to use parentheses if you need to distribute!) What do you do when you multiply or divide an inequality by a negative number? When you multiply or divide an inequality by a negative number, you must flip the inequality sign! 1. 2. 3. 4. 5. Be careful with this one! 6. 7. 8. Thinking ahead In Geometry we will use proportions to find side lengths of similar triangles. If triangle DEF is similar to triangle JKL, what do you think is the length of LK?
Equations of Lines Fill in the missing information for writing equations of lines. Slope Formula: Point-Slope: Slope-Intercept Form: Standard Form: Parallel lines have the SAMEslope! Perpendicular lines have OPPOSITE RECIPROCAL slopes! 1. Find the slope of the line with the given points. 2.Find the slope and equation of each line. a. b. (5, 8) and (0, -2) Undefined Slope 3.Determine if lines AB and CD are parallel, perpendicular, or neither. (Hint: slope!) A(-2, 3) B(2, 6) C(-1, 0) D(3, 3) Parallel SHOW ALL WORK 4. Write the equation of the line that passes through the points (1, 6) and (-2, -6) In slope-intercept form! 5. Graph the equation y = -2x 1 6. Write an equation of a line that passes through the point (1, 4) and is parallel to the line y = 2x + 5. In slope-intercept form! 7. A Write the equation of the line that passes through point A and is perpendicular to the given line.
Systems of Equations In Geometry, systems of equations will appear in almost every unit! You will not be permitted to use the calculator functions to solve the system for you. Geometry teachers at Mercer model the high school courses at John Champe High School. Algebra 2 (with or without Trig) classes at JCHS do not use calculators, so you will need to be able to solve a system by hand. Other than graphing, what are the two methods we use to solve a system of equations? 1. Substitution 2. Elimination Solve the following systems using the most appropriate (or preferred) method. Do not forget to find both variables and show all work! 1. 2. (6, -7) (2, -2) 3. 4. One integer is three more than twice the second integer. The sum of the two integers is -9. Find the numbers. (1, 6) The two intergers are -4 & -5. 5.At Great Wolf Lodge Ski Resort, skis cost $16 to rent and snowboards cost $19. If 28 people rented on a certain day and the resort brought in $478, how many skis and snowboards were rented? 6. 18 skis and 10 snowboards were rented. (10, -6) Thinking ahead In Geometry, we may be given the followingrectangle. Think about how solving systems of equations will help us find the side lengths of the rectangle! 3y 6 6y 6x 6x 12
Factoring and Solving Quadratics In Geometry, factoring will appear in almost every unit! You will not be permitted to use the calculator functions to factor the polynomial for you. Geometry teachers at Mercer model the high school courses at John Champe High School. Algebra 2 (with or without Trig) classes at JCHS do not use calculators, so you will need to be able to solve quadratics by hand. Standard Form of a Quadratic: Quadratic Formula: When is it appropriate to use the square root method to solve a quadratic function? It is appropriate to use the square root method when b = 0, given. Therefore the equation appears as no middle term is present! Solve each of the following quadratics using the most appropriate method. If the quadratic does not factor, you will be instructed to use the quadratic formula. Otherwise, please do not use it. 1. 2. Remember: Always set your equation equal to zero! 3. 4.(Quadratic Formula) 5. 6.
7. 8. 9. 10. Hint: You CANNOT divide by x!! 11. 12. Thinking ahead: In Geometry we may solve a quadratic function by factoring and find that the two solutions are x = 5 and x = -3. We would then ask ourselves, If I substitute my answers back into the problem to find the side lengths of the triangle, will both solutions work?.what do you think?! The perimeter of the triangle is 50 inches. x 2 20 2x + 15
Radicals The very first thing we do in Geometry is find distance using this formula: In most cases, answers will be left in simplest radical form. If there is a radical in the denominator of a fraction, we need to get rid of it by multiplying by a fraction of 1. This process is called rationalizing the denominator! Simplify each radical below. Hint: Use a factor tree to find the factors of each number if necessary! 1. 2. 3. 4. 5. 6. Pay attention to cube root vs. square root! 7. 8. Hint: distribute! Thinking ahead Like we mentioned above, one of the first things we will be learning next year is the distance formula. When we use the distance formula, we keep all non-perfect square answers as radicals in simplest form! Be ready to simplify lots of radicals!!