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Park Forest Math Team Meet #3 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. 2. 3. 4. 5. Mystery: Problem solving Geometry: Properties of Polygons, Pythagorean Theorem Number Theory: Bases, Scientific Notation Arithmetic: Integral Powers (positive, negative, and zero), roots up to the sixth Algebra: Absolute Value, Inequalities in one variable including interpreting line graphs 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! !""#$%&$$$$$$$'()*(+,$-.//$ ! Category 2 ± Geometry !" #$%&'(%)*+*,(-(*%&.)/$0-(%1(-+23%&'(%1/*(%4(/*5.(*%ʹͶ%)$,'(*3%/$6%&'(%'()0'&% 4(/*5.(*%ͷ%LQFKHV+RZPDQ\LQFKHVDUHLQWKHWULDQJOH¶VSHULPHWHU"% % % &$'(&)% % !"#$% % 7" 6SRQJHEREDQG3DWULFNHDFKGUHZDUHJXODUSRO\JRQ3DWULFN¶VSRO\JRQKDG WZLFHDVPDQ\VLGHVDQGVL[WLPHVDVPDQ\GLDJRQDOVDV6SRQJHERE¶V% 8+2%4/$9%*)6(*%&+%:DWULFN¶VSRO\JRQ"% % ;" #$%&'(%6./2)$0%1(-+2<% തതതത%)*%=/./--(-%&+%ܧܥ തതതത % x %ܦܤ തതതത ൌ ͵ܿ݉%3%ܧܦ തതതത ൌ Ͷܿ݉3%/$6%ܦܤ തതതത ൌ ͳͲܿ݉% x ܤܥ 8+2%4/$9%,($&)4(&(.*%*>5/.(6%/.(%)$%&'(%*'/6(6%&./=(?+)6%@ܦܧܥܤ% % ! -! .! Answers 1. _______________ /! 2. _______________ 3. _______________ """#$%&'%#()*! ! ,! +! !""#$%&$$$$$$$'()*(+,$-.//$ ! !"#$%&'#()!&#!*+&,-#./!0!±!1,#2,&,./! Answers 1. 50 2. 12 3. 30 ! 34 5(!,+67!7+$8!#8!&7,!&.'+(-$,9!:,!! ଶ ଶ -! ଶ 7+;,!ͷ ͳʹ ൌ ͳͻ ൌ ! ݁ݏݑ݊݁ݐݕܪ +,! +(<!&7,.,8#.,!&7,!=/>#&,(%),!2,+)%.,)!ͳ͵!'(67,)9!+(<!&7,!:7#$,!>,.'2,&,.! 2,+)%.,)!ͳ͵ ͳ͵ ʹͶ ൌ ͷͲ݄݅݊ܿ݁ݏ4! ! ଵ 04 ,I6SRQJHERE¶VSRO\JRQKDVܰ!)'<,)9!&7,(!'&!7+)! ή ܰ ή ሺܰ െ ͵ሻ!<'+-#(+$)4! ଶ ଵ 3DWULFN¶VSRO\JRQWKHQKDVʹ ή ܰ!)'<,)!+(<! ή ሺʹ ή ܰሻ ή ሺʹ ή ܰ െ ͵ሻ!<'+-#(+$)4! ଶ "#!8.#2!&7,!>.#?$,2!:,!@(#:!&7+&A! ሺʹ ή ܰሻ ή ሺʹ ή ܰ െ ͵ሻ ൌ ή ܰ ή ሺܰ െ ͵ሻ! :7'67!:,!6+(!)'2>$'8/!&#!ሺʹ ή ܰ െ ͵ሻ ൌ ͵ ή ሺܰ െ ͵ሻ4!B7,!)#$%&'#(!')!ܰ ൌ !+(<! VR3DWULFN¶VSRO\JRQKDVʹ ή ܰ ൌ ͳʹ!)'<,)4! ! തതതത!')!>+.+$$,$!&#!ܧܥ തതതത !&7,(!&7,!&.'+(-$,)!!ܦܤܣ+(<!!ܧܥܣ+.,!)'2'$+.4! C4 "'(6,!ܦܤ തതതത തതതത ா ଷ തതതത ଶ തതതത B7,.,8#.,!തതതത ൌ തതതത ൌ !+(<!)'(6,!'(!&.'+(-$,!!ܦܤܣ:,!7+;,!ܤܣ ܦܣଶ ൌ ͳͲଶ ! ସ തതതത ൌ ܿ݉!+(<!ܦܣ തതതത ൌ ͺܿ݉4!B7,.,8#.,!&7,!+.,+!#8!&.'+(-$,!!)'!ܦܤܣ :,!-,&!&7+&!ܤܣ ʹͶܿ݉ଶ+(<!&7,!+.,+!#8!&.'+(-$,!!)'!ܧܥܣͷͶܿ݉ଶ9!+(<!&7,!&.+>,D#'<!')!&7,'.! <'88,.,(6,9!͵Ͳܿ݉ଶ 4! """#$%&'%#()*! ! Category 2 Geometry Meet #3, January 2009 1. How many degrees are in the sum of the interior angles of a convex decagon? 2. Let the number of diagonals in a regular octagon be !, and the number of diagonals in a regular hexagon be ". What is the value of ! # " ? 3. Quadrilateral ABCD has right angles at A and C. The lengths of CD, BC, and AB are 7 cm, 11cm, and 1cm respectively. How many centimeters long is AD? C B A D Answers 1. _______________ 2. _______________ 3. _______________ Solutions to Category 2 Geometry Meet #3, January 2009 Answers 1. 1440 2. 11 3. 13 1. For an !-sided convex polygon, the expression "#$%! & '( will give the number of degrees in the sum of the interior angles of the polygon. So a decagon %! ) "$( would have "#$%"$ & '( ) "#$%#( ) "**$ degrees. +%+,-( 2. For an !-sided convex polygon, the expression will . give the number of diagonals in the polygon. So an octagon has /%0( 1%-( ) '$ diagonals and a hexagon has ) 2 diagonals. So . . 3 ) '$ and 4 ) 2. Therefore 3 & 4 ) '$ & 2 ) "". 3. First draw in segment 56. Since 56 is the hypotenuse of triangle 576, we can use the Pythagorean Theorem to find the length of drawn in segment 56. 56. ) 8. 9 "". ) *2 9 "'" ) "8$. However BD is also the hypotenuse of triangle :56 and therefore 56. ) ". 9 :6. ) " 9 :6. ) "8$. So :6. ) ";2 and :6< ) <"=. 11 C B 7 1 A ? D Category 2 Geometry Meet #3, January 2007 1. How many degrees are in the measure of an exterior angle of a regular decagon? exterior angle 2. A Pythagorean Triple is a set of three natural numbers that satisfy the Pythagorean Theorem a 2 + b 2 = c 2 , where a and b are legs on a right triangle and c is the hypotenuse. One way to find Pythagorean Triples is with the three equations a = m 2 − n 2 , b = 2mn , and c = m 2 + n 2 , where the values of m and n are natural numbers with m > n. How many units are in the perimeter of the right triangle that it is produced when m = 7 and n = 4? 3. An isosceles triangle has sides measuring 34 units, 34 units, and 32 units. If the 32-unit side is considered the base, how many units are in the height (or the altitude) of this triangle? Answers 1. _______________ 2. _______________ 3. _______________ 34 units 34 units 32 units www.imlem.org Solutions to Category 2 Geometry Meet #3, January 2007 Answers 1. 36 2. 154 1. Imagine taking a walk counter-clockwise around the regular decagon. You will make ten left-hand turns of the same measure. When you get back to where you started, you will have turned a total of 360 degrees. Therefore, each exterior angle must be 360 ÷ 10 = 36 degrees. 3. 30 2. Using m = 7 and n = 4, we find that a = 7 2 − 4 2 = 49 −16 = 33 , b = 2 ⋅ 7 ⋅ 4 = 56 , and c = 7 2 + 4 2 = 49 + 16 = 65 . The perimeter of a right triangle with sides of 33, 56, and 65 units is 33 + 56 + 65 = 154 units. 3. First we draw the height we wish to find. An altitude line is always perpendicular to the base. On an isosceles triangle this line also meets the base at the midpoint. Now we can use the Pythagorean Theorem to find the missing leg of a right triangle with an hypotenuse of 34 units and a leg of 16 units. 342 = 1156 and 162 = 256. 1156 – 256 = 900 and 900 = 30 , so the height of the isosceles triangle must be 30 units. 34 units 34 units 16 units and 16 units www.imlem.org Category 2 Geometry Meet #3, January 2005 1. A certain polygon has twice as many diagonals as sides. How many sides are there on this polygon? Note: A diagonal in a polygon is any line segment that connects two vertices and is not a side. 2. The figure below shows the design of a raft which floats in the middle of a lake. The raft is made of a number of square sections that are linked together. If the perimeter of the raft is 220 feet, how many square feet are in the area of the raft? 3. In the figure below, triangles ABC, ACD, and ADE are right triangles, and sides AB, BC, CD, and DE have the same measure. If the measure of side AB is 2 centimeters, how many centimeters are there in the measure of side AE? E D C Answers 1. _______________ 2. _______________ 3. _______________ B www.Imlem.org A Solutions to Category 2 Geometry Meet #3, January 2005 Average team got 18.63 points, or 1.55 questions correct 1. The table below shows the number of diagonals for several polygons. The heptagon with 7 sides has 14 diagonals, which is twice the number of sides. Answers 1. 7 Polygon Triangle Square Pentagon Hexagon Heptagon Octagon 2. 625 3. 4 Sides 3 4 5 6 7 8 Diagonals 0 2 5 9 14 20 2. There are 44 side lengths of the square sections in the perimeter of the raft, so each square must have a side length of 220 feet ÷ 44 = 5 feet. The area of each square is thus 5 feet × 5 feet = 25 square feet. The raft is made of 25 sections, so the area of the raft is 25 × 25 square feet = 625 square feet. E D C 3. We will have to use the Pythagorean Theorem three times to calculate the length of each hypotenuse. Let the length of AC = x, the length of AD = y, and the length of AE = z. Then we find x as follows: x2 = 22 + 22 x2 = 8 x = 8 x = 2 2 We find y as follows: y2 = 22 + B A ( 8) 2 y 2 = 12 y = 12 y = 2 3 Finally, we find z as follows: z 2 = 22 + ( 12 ) 2 z 2 = 16 z = 4 . AE is 4 centimeters. www.Imlem.org So the length of