Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Duality (projective geometry) wikipedia , lookup
Multilateration wikipedia , lookup
History of trigonometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Euler angles wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Integer triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Rational trigonometry wikipedia , lookup
GEOMETRY HONORS SUMMER ASSIGNMENT June 2016 Dear Geometry Honors Students, Welcome to the beginning of your journey through the Honors Mathematics Program at Randolph High School. Geometry is the beginning of this exciting adventure. This coming school year, you will build a solid foundation that focuses on your ability to use critical thinking and reasoning skills. This assignment is designed to prepare you for an easy transition into our program. Although the Homework Administrative Regulations of the Randolph Township Schools do not permit us to “grade” your work on this summer assignment, the material in this packet will be the focus of instruction during the first few days of school and you will be tested on this material within the first week of school. The material contained in this packet is not new - it is preparatory and review for the ensuing school year. There are three parts to this assignment. The first and second parts ask you to build on your Geometry vocabulary and apply your knowledge. The third part is a review of the Algebra One concepts you will be expected to know and be able to use when you solve Geometry problems. As you work through this assignment, feel free to contact me with any questions or concerns via email at [email protected]. I look forward to meeting all of you in September. Teresa Schuele Mathematics Teacher Class of 2018 Advisor Randolph High School PART l: Basic Terminology Directions: Find the term that best represents the definition given below. 1. GEOMETRY The study of the properties and the relationships of points, lines, planes, and solids 2. An unidentified term represented as a dot on a piece of paper; usually named by a capital letter; has no length, width, of thickness; merely indicates a position. 3. A set of points that may form a straight or curved line 4. A set of points that forms a completely flat surface extending indefinitely in all directions. LINES AND LINE SEGMENTS 5. A set of points all of which lie on the same straight line 6. A set of three or more points that are not found on the same straight line 7. A set points consisting of two points on a line, called endpoints, and all points on the line between the endpoints. 8. Segments that have the same length 9. The point of that line segment that divides the segment into two congruent segments 10. Any line or subset of a line that intersects the segment at its midpoint 11. A part of a line that consist of a point on the line, called an endpoint, and all the points on one side of the endpoint. 12. Two rays of the same line with a common endpoint and no other points in common ANGLES 13. A set of points that is the union of the two rays having the same endpoints 14. An angle that is the union of opposite rays whose measure is 180 15. An angle whose measure is 90 16. An angle whose measure is greater than 0 and less than 90 17. An angle whose measure is greater than 90 and less than 180 18. An angle whose measure is greater than 180 and less than 360 19. Angles that have the same measure 20. A ray whose endpoint is the vertex of the angle and which divides the angle into two congruent angles PAIRS OF ANGLES 21. Two angles in the same plane that have a common vertex and a common side, but do not have any interior points in common 22. Two angles in which the sides of one angle are opposite rays to the sides of the second angle 23. Two angles the sum of whose measure is 90 degrees 24. Two angles the sum of whose measure is 180 degrees 25. Two angles that have a common side and their remaining sides are opposite rays LINES 26. Two lines that intersect to form right angles 27. Two lines that never intersect 28. A line, line segment, or ray that is perpendicular to a line segment and bisects that line segment TRIANGLES AND LINE SEGMENTS ASSOCIATED WITH TRIANGLES 29. A closed figure in a plane that is the union of line segments such that the segments intersect only at their endpoints and no segments sharing a common endpoint are collinear 30. A polygon that has exactly three sides 31. A triangle that has two congruent sides 32. A triangle that has three congruent sides 33. A triangle that has three acute angles 34. A triangle that has three congruent angles 35. A triangle that has a right angle 36. A triangle that has an obtuse angle 37. A triangle that has no congruent sides 38. A line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side 39. A line segment drawn from any vertex of the triangle, ending at the midpoint of the opposite side PART ll: Application 1. BD divides the right <ABC into two parts. The ratio of the measures of <ABD to the measure of <DBC is 3:2. Find the measure of <ABD. A D C B A 2. AB AC , and AB is 3 times as long as BC. If the perimeter of triangle ABC is 28, find AB. B C S 3. Solve for y in terms of x. m<PQS = x + y m<SQR = 3x – 8 P Q R 4. A point P is randomly chosen on AB. What is the probability that it is within 5 units of C? A C B -6 -3 10 5. The measure of the supplement of an angle is 30 more than four times the measure of the complement of the angle. Find the measure of the complement. A C 6. If the m<1 = 6x2 + 15x and the m<2 = 4x + 10, find the m<EBD. 1 B 2 E D 7. m<K = 3x – 42 and <K is obtuse. Find the restriction(s) on the value of x. 8. Given points F, G, and H. FG = 18, FH = 7, and GH = 11. Describe the relative positions of F, G, and H. Justify your conclusion. 9. Find the sum of x and y. 4y+14 3x+7y F 10. The perimeter of triangle PQR is at least 50. If y > 7, what do you know about the value of x? 22 4x-2y G 18 H 11. Answer each of the following questions. Be sure to completely justify your answer. a. If AB BC, does this imply B is the midpoint of AC? b. Is it possible for an obtuse angle to be complementary to an acute angle? c. Can a right angle be one of two supplementary angles? 12. <ABC is a right angle. BD and BE trisect <ABC. 3 m<ABD = 3x y Solve for x and y. 2 8x y 2 m<DBE = 3 9 y 2x m<EBC = 2 A D E B C Part lll: Algebra Review 1. Solve the following word problems by: a. drawing a diagram b. writing an equation c. solving the equation. a. Separate 150 into two parts such that four times the larger exceeds five times the smaller by 60. b. The length of a rectangle exceeds its width by four feet. If the width is doubled and the length is diminished by two feet, a new rectangle is formed whose perimeter is eight feet more than the perimeter of the original rectangle. Find the dimensions of the original rectangle. 2. Simplify: x y x ( x y )2 a. 2 x y2 x y x4 y4 b. x 2 2 xy 8 y 2 5x 10 y x 2 16 y 2 3x 12 y 3. Solve for x. a. ax bx 4a 4b b. 4. Simplify the radicals. a. 5 18 y 3 w 6 z 8 b. x 8 y 3 2 x 2 y c. a b 5. Solve the inequality: a. 90 < 5x + 5 < 180 6. Solve for x. a. 5y2 – 1 = 2y 1 1 1 x c d b. a b d. 60 3 8 3t 4 2t 4 5t 1 3 6 9 b. (x + 4)2 = (x – 4)2 + 96 7. Review all methods of factoring & solving – simple, AC method, completing the square, quadratic formula, special patterns – difference of squares and perfect square trinomials. Write and solve an example of each. 8. Review solving systems of linear equations – graphing, elimination, substitution. Write and solve an example of each.