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TEAMS 9 Summer Assignment2.Pdf Geometry Geometry Honors T.E.A.M.S. Geometry Honors Summer Assignment 1 | P a g e Dear Parents and Students: All students entering Geometry or Geometry Honors are required to complete this assignment. This assignment is a review of essential topics to strengthen math skills for the upcoming school year. If you need assistance with any of the topics included in this assignment, we strongly recommend that you to use the following resource: http://www.khanacademy.org/. If you would like additional practice with any topic in this assignment visit: http://www.math- drills.com. Below are the POLICIES of the summer assignment: The summer assignment is due the first day of class. On the first day of class, teachers will collect the summer assignment. Any student who does not have the assignment will be given one by the teacher. Late projects will lose 10 points each day. Summer assignments will be graded as a quiz. This quiz grade will consist of 20% completion and 80% accuracy. Completion is defined as having all work shown in the space provided to receive full credit, and a parent/guardian signature. Any student who registers as a new attendee of Teaneck High School after August 15th will have one extra week to complete the summer assignment. Summer assignments are available on the district website and available in the THS guidance office. HAVE A GREAT SUMMER! 2 | P a g e An Introduction to the Basics of Geometry Directions: Read through the definitions and examples given in each section, then complete the practice questions, found on pages 20 to 26. Those pages will be collected by your Geometry Teacher on the first day of school. Section 1: Points, Lines and Planes Undefined term: words that do not have formal definitions, but there is an agreement about what they mean. In Geometry, the words point, line and plane are undefined terms. Undefined Term Meaning Example/Picture and symbols A point has no dimension but has location. A dot is Point used to represent a point. A line has one dimension. It is represented by a line with two arrowheads, showing that it extends in two directions without end. Line Through any two points there is exactly one line. You can use any two points on a line to name it, or it can be named by a lowercase letter written by the line. A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it extends without end. Plane Through any three points not on the same line, there is exactly one plane. You can use three points that are not on the same line to name a plane, or you can use a capital letter (without a point next to it) to name a plane. Collinear points: points that lie on the same line. Coplanar points: points that lie in the same plane. Example 1: Naming Points, Lines and Planes a. Give two other names for 푃푄⃡ and plane R. Answer: Other names for 푃푄⃡ are 푄푃⃡ and line n. Other names for plane R are plane SVT and plane PTV. b. Name three points that are collinear. Name four points that are coplanar. Answer: Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so they are coplanar. 3 | P a g e Defined Terms: Segment and Ray The definitions below use line AB (written as 퐴퐵⃡ ) and points A and B. Defined Term Definition Example/Picture and Symbols The line segment AB, or segment AB (written as 퐴퐵̅̅̅̅) consists of the endpoints A and B and all points on 퐴퐵⃡ that are between A and B. The endpoints are like stop and start points. Unlike lines, segments do not continue on forever in both Segment directions and they can be measured. 퐴퐵̅̅̅̅ (read “segment AB”) Note that 퐴퐵̅̅̅̅ can also be called 퐵퐴̅̅̅̅. The ray AB (written as 퐴퐵 ) consists of the endpoint A and all points on 퐴퐵⃡ that lie on the same side of A as B. In other words, rays have a starting point (called an endpoint) and Ray continue in the direction of the other point. Note that 퐴퐵 and 퐵퐴 are two different rays because they are Top: 퐴퐵 (read “ray AB”) going in different directions. Bottom: 퐵퐴 (read “ray BA”) If point C lies on 퐴퐵⃡ between A and B, then 퐶퐴 and 퐶퐵 are opposite rays. Opposite Rays They have the same point but go in opposite directions to form 퐶퐴 and 퐶퐵 are opposite rays. a line. Two or more geometric figures intersect when they have one or more points in common. The intersection of the figures is the set of all points they have in common. Intersection 4 | P a g e Example 2: Naming Segments, Rays and Opposite Rays a. Give another name for 퐺퐻̅̅̅̅ . Answer: Another name for 퐺퐻̅̅̅̅ is 퐻퐺̅̅̅̅. b. Name all rays with endpoint 퐽. Which of these rays are opposite rays? Answer: The rays with endpoint 퐽 are 퐽퐸 , 퐽퐺 , 퐽퐹 , and 퐽퐻 . The pairs of opposite rays with endpoint 퐽 are 퐽퐸 and 퐽퐹 , and 퐽퐺 and 퐽퐻 . Practice Questions for Section 1 can be found on Page 20. Section 2: Measuring Segments In Geometry, a rule that is accepted without proof is called a postulate or an axiom. A rule that can be proved is called a theorem. The Ruler Postulate: The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB (notice there is no symbol above the 2 letters), is the absolute value of the difference of the coordinates of A and B. Congruent Segments: Line segments that have the same length are called congruent segments. You can say “the length of 퐴퐵̅̅̅̅ is equal to the length of 퐶퐷̅̅̅̅,” or you can say “퐴퐵̅̅̅̅ is congruent to 퐶퐷̅̅̅̅.” The symbol ≅ means “is congruent to.” In the diagram above, of 퐴퐵̅̅̅̅ and 퐶퐷̅̅̅̅ have tick marks on them, indicating 퐴퐵̅̅̅̅ ≅ 퐶퐷̅̅̅̅. When there is more than one pair of congruent segments, use multiple tick marks. 5 | P a g e When three points are collinear, you can say that one point is between the other two. Segment Addition Postulate If B is between A and C, then AB + BC=AC. If AB+BC=AC, then B is between A and C. Example 1: Comparing Segments for Congruence a. Plot J(−3, 4), K(2, 4), L(1, 3), and M(1, −2) in a coordinate plane. Then determine whether ̅퐽퐾̅̅ and 퐿푀̅̅̅̅ are congruent. Answer: Plot the points, as shown. To find the length of a horizontal segment, find the absolute value of the difference of the x-coordinates of the endpoints. 퐽퐾 = |−3 − 2| = 5, Ruler Postulate To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. 퐿푀 = |3 − (−2)| = 5, Ruler Postulate 퐽퐾 = 퐿푀. So, ̅퐽퐾̅̅ ≅ 퐿푀̅̅̅̅. Example 2: Using the Segment Addition Postulate a. Find 퐷퐹. Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 퐷퐹. 6 | P a g e b. Find 퐺퐻. Answer: Use the Segment Addition Postulate to write an equation. Then solve the equation to find 퐺퐻. Example 3: Using the Segment Addition Postulate The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri. Answer: 1. Understand the Problem. You are given the distance from Lubbock to St. Louis and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis. 2. Make a Plan. Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis. 3. Solve the Problem. Use the Segment Addition Postulate to write an equation. Then solve the equation to find 푇푆. So, the distance from Tulsa to St. Louis is 361 miles. Practice Questions for Section 2 can be found on Page 21. 7 | P a g e Section 3: Using Midpoint and Distance Formulas Midpoints and Segment Bisectors The midpoint of a segment is the point that divides the segment into two congruent segments. A segment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint. A midpoint or a segment bisector bisects a segment. (“Bi” means two, “sect” means sections). Example 1: Finding Segment Lengths In the skateboard design, 푉푊̅̅̅̅̅ bisects 푋푌̅̅̅̅ at point T, and 푋푇 = 39.9 cm. Find 푋푌. Answer: Point 푇 is the midpoint of 푋푌̅̅̅̅, so 푋푇 = 푇푌 = 39.9 cm. 푋푌 = 푋푇 + 푇푌 Segment Addition Postulate 푋푌 = 39.9 + 39.9 Substitute 푋푌 = 79.8 Add So, 푋푌 = 79.8 cm 8 | P a g e Example 2: Using Algebra with Segment Lengths Point M is the midpoint of 푉푊̅̅̅̅̅. Find the length of 푉푀̅̅̅̅̅. Answer: 1. Write and solve an equation. Use the fact that 푉푀̅̅̅̅̅ = ̅푀푊̅̅̅̅̅. 푉푀 = 푀푊 Write the equation 4푥 − 1 = 3푥 + 3 Substitute 푥 − 1 = 3 Subtract 3푥 from both sides 푥 = 4 Add 1 to each side 2. Evaluate 푉푀 = 4푥 − 1 when 푥 = 4 푉푀 = 4(4) − 1 = 15 So the length of 푉푀̅̅̅̅̅ is 15 units. Using the Midpoint Formula The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates of the endpoints.
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