Geometrygeometry
Total Page:16
File Type:pdf, Size:1020Kb
Park Forest Math Team Meet #3 GeometryGeometry Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics 5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities Important Information you need to know about GEOMETRY… Properties of Polygons, Pythagorean Theorem Formulas for Polygons where n means the number of sides: • Exterior Angle Measurement of a Regular Polygon: 360÷n • Sum of Interior Angles: 180(n – 2) • Interior Angle Measurement of a regular polygon: • An interior angle and an exterior angle of a regular polygon always add up to 180° Interior angle Exterior angle Diagonals of a Polygon where n stands for the number of vertices (which is equal to the number of sides): • • A diagonal is a segment that connects one vertex of a polygon to another vertex that is not directly next to it. The dashed lines represent some of the diagonals of this pentagon. Pythagorean Theorem • a2 + b2 = c2 • a and b are the legs of the triangle and c is the hypotenuse (the side opposite the right angle) c a b • Common Right triangles are ones with sides 3, 4, 5, with sides 5, 12, 13, with sides 7, 24, 25, and multiples thereof—Memorize these! Category 2 50th anniversary edition Geometry 26 Y Meet #3 - January, 2014 W 1) How many cm long is segment 6 XY ? All measurements are in centimeters (cm). Z X 8 C 2) Angle ABC is a right X angle. Triangle BCD is E an isosceles triangle D such that DB = BC. 120 150 Find the value of X if it is the measure in degrees of angle 70 EDC and X < 180. B A 3) Moe and Larry race from point B to point S at a rectangular field. Moe runs from B to A to S at an average rate of 5 feet every second. Larry runs diagonally across the field from B to S at an average rate of 10 feet every 3 seconds. If they both leave point B at the same time, then who wins the race? Also, by how many seconds does the winner finish ahead of the runner-up? (You must answer both questions correctly to receive credit.) E S ANSWERS 1) __________ cm 50 feet 2) __________ B 120 feet A 3) __________ winner _________ seconds www.imlem.org Solutions to Category 2 Geometry Meet #3 - January, 2014 Answers 1) Use the Pythagorean Theorem twice - first to find the length of WX and then XY. 1) 24 2) 155 3) Moe Use this result to find XY: 5 2) The measure of angle DBA is 20 degrees, because the sum of the angles of a quadrilateral is 360 degrees. The measure of angle DBC is 70 degrees, because angle ABC is a right angle (90 degrees). Since two sides (DB and BC) of triangle DBC are congruent, the angles opposite those sides are congruent. The vertex angle, DBC, measures 70 degrees, so the base angles are 55 degrees each, including angle BDC. angle X + 150 + 55 = 360, so X = 155. 3) Both answers must be answered correctly in order for students to receive credit. Moe: runs 120 + 50, or 170 feet. At a rate of 5 feet per second, it takes him 170 / 5, or 34 seconds to reach point S. Larry: Use the Pythagorean Theorem to find that he has run 130 feet. At a rate of 10 feet every 3 seconds, it takes him (130 / 10) x 3, or 39 seconds to reach point S. Moe, therefore, reaches point S ahead of Larry by 39 - 34, or by 5 seconds, so Moe wins the race. www.imlem.org Meet #3 January 2012 Category 2 – Geometry (11, 4) 1. Given the coordinates in the diagram, what is the distance between the two points? (-1, -1) 2. How many diagonals are there in a regular polygon with sides (a Hexadecagon)? 3. The sum of interior angles in a regular polygon is times as great as the measure of each of its exterior angles. How many sides does the polygon have? Answers 1. __________ Units 2. _________ Diagonals 3. __________ Sides www.imlem.org Meet #3 January 2012 Solutions to Category 2 – Geometery Answers 1. 1. The horizontal distance is units, and the vertical distance is 2. units, so the total distance is √ units. 3. 2. The formula for the number of diagonals in a polygon with N sides is: so in our case we’ll have diagonals. 3. The exterior angles of a polygon all add up to 360 degrees, so if there are sides to the polygon, then each exterior angle measures degrees. Every interior angle measures degrees, and their sum is therefore degrees. So in our case we’re told that: whice we can rewrite as: . Though this is technically a quadratic equation, we know that is a natural number and can easily find that is a solution (an Octagon). [The other solution, , is clearly not an answer to our problem]. www.imlem.org Category 2 - Geometry Meet #3, January 2010 1. The number of diagonals in a polygon is four times the number of its vertices. How many vertices does it have? (A diagonal is a line segment that connects two non-adjacent vertices of a polygon). 2. The exterior angle to a regular polygon N (with N sides) is half that of a regular polygon M (with M sides). Polygon N has 7 times as many diagonals as polygon M. What is the value of 푀 ∙ 푁? 3. Tom stands exactly 2 miles west of Jerry. At 10:00am Tom starts walking east at 5 mph (miles per hour). At 10:20am Jerry starts heading north at 9 mph. How many miles between them at 12:00pm (noon)? Answers 1. _______________ 2. _______________ Remember: You do not have to specify units. Specifying 3. _______________ the wrong units will disqualify your answer. www.imlem.org Solutions to Category 2 - Geometry Answers Meet #3, January 2010 1. 11 2. 50 3. 17 1. If we call the number of vertices V, then the number of diagonals is given by the 푉×(푉−3) 푉−3 formula: and for this to equal 4 × 푉 we get = 4 or 푉 = 11. 2 2 360 2. The exterior angle is so it should be clear that 푁 = 2푀. 푁푢푚푏푒푟 표푓 푠푑푒푠 푀∙(푀−3) 푁∙(푁−3) 2푀∙(2푀−3) Polygon M will have diagonals, and polygon N will have = 2 2 2 diagonals. So we need to solve: 2푀 ∙ 2 ∙ 푀 − 3 = 7 ∙ 푀 ∙ 푀 − 3 which we can simplify to 4 ∙ 푀 − 6 = 7푀 − 21 and then to 3 ∙ 푀 = 15 and the solution is 푀 = 5, 푁 = 10 and 푀 ∙ 푁 = 50. 3. At 12:00pm, having walked for two hours, Tom is 10 miles to the east of his original location, which means he’s 8 miles east of Jerry’s original location. Jerry, after running at 9 mph for 1 hour and 40 minutes is 15 miles north of his (own) original location. The distance between them is 82 + 152 = 289 = 17 푚푙푒푠. Note that we have to compare their locations to some ‘fixed’ or agreed-upon point. You could have used any other point-of-reference and get the same answer. www.imlem.org Category 2 Geometry Meet #3, January 2008 1. If the exterior angle of a regular polygon is 24 degrees, how many sides does the polygon have? 2. In the figure to the right, four of the sides of a regular polygon are shown. Also drawn are all of the diagonals that use A as one of the endpoints of the diagonal; however the full lengths of the diagonals are not always shown. How many diagonals does this polygon have? A 3. In the figure below, triangle ABC is a right triangle with right angle at A and quadrilateral BCDE is a square. If AB = 8 and AC = 18 , what is the area of the pentagon ABEDC? B C A E Answers 1. _______________ D 2. _______________ 3. _______________ Solutions to Category 2 Geometry Meet #3, January 2008 Answers 1. Since the sum of the exterior angles is always 360 degrees, if you divide 360 degrees by 24 degrees you get 15 which is the 1. 15 number of equal exterior angles and therefore 15 vertices and sides to the polygon. 2. 44 3. 20 2. If there are 8 diagonals and only 2 of the diagonals’ other endpoints are shown there must be 6 vertices that are not shown. Since 5 vertices are shown there are a total of 11 vertices and 11 sides. If there are 11 sides there are ͥͥʚͥͥͯͧʛ diagonals total. ͦ Ɣ ̍̍ 3. Using the Pythagorean Theorem: → ͦ ͦ → ͦ ͦ ͦ ͦ ͦ Since̼̽ŬŬŬŬ Ɣ Ŭ ̻̼ŬŬŬ isƍ the ̻̽ŬŬŬ Ŭarea of the̼̽ŬŬŬŬ square,Ɣ √8 theƍ area√18 of square̼̽ŬŬŬŬ EBCDƔ 8ƍ18 is 26. Ɣ The 26 pentagon ͦ we are̼̽Ŭ ŬlookingŬŬ for though is equal to the area of the square minus the area of the right triangle ABC. The area of triangle ABC = √ͬ√ͥͬ √ͥͨͨ ͥͦ . The area of pentagon ABEDC = 26 – 6 = 20 . ͦ Ɣ ͦ Ɣ ͦ Ɣ 6 B C A E D Category 2 Geometry Meet #3, January 2006 1. How many degrees are in the measure of an interior angle of a regular 15-gon? 1 2.