1. Sketch a concave polygon and explain why it is both concave and a polygon. A polygon is a simple closed curve that is the union of line segments.

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1 SOLUTIONS MATH / Fial Review Questios, F5. Sketch a cocave polygo ad explai why it is both cocave ad a polygo. A polygo is a simple closed curve that is the uio of lie segmets. A polygo is cocave if it cotais two poits such that the lie segmet joiig the poits does ot completely lie i the polygo.. Sketch a simple closed curve ad explai why it is both simple ad closed. A simple closed curve is a simple curve (starts ad stops without itersectig itself) that starts ad stops at the same poit.. Determie the measure of the vertex, cetral ad exterior agles of a regular heptago, octago ad oago. Shape Vertex ( )80 Cetral 60 Exterior ( ) regular heptago regular octago regular oago Determie whether each figure is a regular polygo. If it is, explai why. If it is ot, state which coditio it dos ot satisfy. No, sides are ot cogruet No, agels are ot cogruet Yes, all agles are cogruet ad all agles are cogruet 5. A prism has 69 edges. How may vertices ad faces does it have? E = 69, V = 46, F= 5 6. A pyramid has 68 edges. How may vertices ad faces does it have? E = 68, V = 5, F= 5 7. A prism has 4 faces. How may edges ad vertices does it have? E = 66, V = 44, F= 4

2 8. A pyramid has 4 faces. How may edges ad vertices does it have? E = 46, V = 4, F= 4 9. Which regular polygos tessellate? Explai ad illustrate your aswer with a sketch. Oly the triagle, square ad hexago will tessellate sice they are the oly regular polygos whose vertex agle divides evely ito Sketch a patter block figure with exactly four rotatioal symmetries ad o lies of symmetry. List the agles of rotatio. 90, 89, 70, 60 Other figures also work. Sketch a patter block figure that is ot a regular polygo, with exactly three lies of symmetry. Mark the lies ad label them L, L ad L. What are these ets for? Be exact: Other figures also work Right, regular petagoal prism Tetrahedro (Right, equilateral pyramid) Octahedro Right, regular, hexagoal pyramid Right cylider. I the followig figure, lie l is parallel to lie m. Give the agle measures idicated o the figure, fid the measures of al of the agles. Use mathematics, ot a protractor. Use cogruet vertical agles ad cogruet alterate iterior agles

3 4. If you halve the legth of each side of a rectagle, by what factor does the PERIMETER of the rectagle expad or cotract? What if you alter the legth by a factor of A? cotracts; A expads if A, A cotracts if A 5. If you double the legth of each side of a rectagular prism, by what factor does the SURFACE AREA of the prism expad or cotract? What if you alter the legth by a factor of A? expads; A expads if A, A cotracts if A 6. If you double the radius of a sphere, by what factor does the VOLUME of the sphere expad or cotract? What if you alter the legth of the radius by a factor of A? expads; A expads if A, A cotracts if A 7. How may CUBIC FEET of gravel are eeded to cover a yard x yard area filled to 0.9 yard high? ft.9 yd.9 7 ft 45.8 ft yd 8. If five teis balls of diameter 6.5 cm. are packed ito a cylidrical ca that exactly holds the five balls (the ca also has diameter 6.5 cm.); what is the volume of the UNUSED SPACE i the ca? cm 9. Modify all three sides of a equilateral triagle to create a shape that tessellates. Briefly explai your work ad use tracig paper to aid you as you sketch the begiig of a tessellatio usig this template. What type of tessellatio is it?. midpoit rotate curve o this side. create curve rotate to ext side Rotatio tessellatio

4 0. Fid the surface area ad volume of the followig objects. Surface Area 8 4 bases frot/back faces 58 0 bottom face cm h h Volume cm hexagoal base faces cm 4. H H cm. The followig equilateral triagle has three circular regios surroudig it. What is the area of the shaded portio of the diagram? mab =.9 iches CD =.75 iches B D A C h 4 h cm triagle cm three 60portios make / circle i cm. What is the area of the regular heptago pictured above? Show your work. Iclude uits.. Lisa is buyig special hardwood floorig from Germay ad the cost (i US dollars) is $500 per budle. Each budle will cover square meters. If Lisa wats to put the floorig i a room that is 8 feet by 0 feet, how much will the floorig cost? Note: Lisa ca oly buy whole budles (e.g. she ca t buy / of a budle, etc). Show all work ad coversios. Note yard.95 meters. yd yd.95 m.95 m 8' 0'.489 m ft ft yd yd budle.489 m.79 budles Lisa must by budles for $4500 m

5 4. This package is iches loger tha it is wide. It is also 6 iches wider tha it is high. If the total legth of the ribbo eeded to go aroud both ways is 96 iches log, what is the legth of the package? l w w 6 h Ribbo covers llww4hlw4h lw4h( w) w4h44w4h 4 4(6 h) 4h4 4 8h96 8h48 h6 iches, w iches, l 4 iches 5. The followig two figures are similar. Each dimesio of the larger figure is TRIPLE the correspodig dimesio of the smaller figure. If this triplig of dimesios is cotiued, how may cubes will there be i the FOURTH (th) figure? If the dimesios of the first figure are doubled rather tha tripled, how may cubes would there be i the SECOND (th) figure? Triplig dimesios sequece Doublig dimesios sequece Figure Volume Volume = 7 = 8 = 8 = = 7 = = 8 = = 7 = 59, = 8 = Alterate solutio Triplig dimesios sequece Doublig dimesios sequece Figure Volume Volume = 8 = 4 ( ) = 6 = 87 ( ) = 6 = 9 4 ( 6 ) = 9 = 59,049 ( 6 ) = 9 = Note that 7 - = - ad 8 - = - so both patters are equivalet 6. Julia ad James just bought their first house. They are so excited they had a scale model of their ew house made. The scale factor relatig their model to the actual house is 75. If the legth of oe side of the actual house is 50 feet, how may iches log is the correspodig side of the model house? If the volume of the model of the house is 0.0 feet, what is the volume of the actual house? iches 50 feet 8 iches foot 75 iches.feet 75 7,900,000 iches foot

6 7. Sketch the traslated image of POLY for the mappig that maps A to B. 8. Sketch the rotated image of BOB for a 90 couterclockwise rotatio about poit A: 9. Sketch the reflected image of ALLY over the lie L 0. Use the motio; poit A to poit B ad lie L to sketch a glide reflectio of the triagle/ POLY BOB A. I the diagram below triagle A B C is the image of triagle ABC uder a rotatio through 90 clockwise cetered at the origi. What are the coordiates of A, B ad C? A = (, -), B = (-, -) ad C = ( 0, -5) Geeralize the rule for this rotatio, i.e., create a rule that would move ay poit the same way. (x, y) is moved to the poit (y, -x)

7 . The followig figure is made usig two ta parallelograms ad oe red trapezoid from the patter blocks. Sketch a elargemet of the figure by a scale factor of. Use as may ta parallelograms ad red trapezoids i your elargemet as possible. If istead you were to sketch a elargemet by a scale factor of (or 4 or 5), how may ta parallelograms ad how may red trapezoids (or combied blocks of the same area) would be i your figure? N Image ( oly) Parallelograms Trapezoid (equivalet to) Match the triagles ito CONGRUENT or (oly) SIMILAR pairs. I each case, solve for ukow values i the triagles (oly as eeded) ad use triagle properties to show why the matched triagle pair is CONGRUENT or SIMILAR. Explai as eeded. Show your work. Triagles are ot to scale. PQR XOF ASA ABC RST SSS R F P X Q O RP FX TS CB TR CA RS AB Pythagorea Thm Triagle sum agles 4 cm A C cm B ABC RST by SSS or SAS or ASA DE DF WV UV ad DF WU Pythagorea theorem W U DF isosceles right triagles E T 4 cm W 5 cm R cm V S R 55 4 cm 0 Q P F cm ABC RST ASA XY CA ZY TA Y A Z O C X D 45 cm 45 F cm U X cm 65.5 cm Y cm 65 A T.5 cm

8 4. I the followig diagram a cliometer was used to determie that the agle of elevatio, agle PEF, from the observers eye to the top of the statue is. This kowledge eables us to correctly draw the scale model of triagle PEF which is similar to the actual triagle formed by the observer s eye ad the two parts of the statue as show i the picture. I our scale triagle, the legth of side EP is 9 iches ad the legth of side EF is 0.8 iches. Suppose both the observer ad the statue are at groud level, the observer is 40 feet from the statue ad the observer s eye is 5 feet above groud. What is the height of the statue? 0.8 EF E ' F ' E ' F ' 9 EP E ' P ' EF ' ' 0.8 EF ' ' The statue is about 5 feet high 5. Every equilateral triagle is isosceles. True by defiitio 6. A rhombus is a regular polygo. False, agles are ot cogruet 7. Every hexago is a regular polygo. False, draw a o regular hexago 8. The acute agles of a right triagle are complemetary. True, agle sum: ABC 80 C 90 AB90 9. The acute agles of a right triagle are supplemetary. False, see If two agles are supplemetary, oe must be either right or obtuse. True, supplemetary meas they add to 80. If they are both < 90, they ca t add to Every prism has a square base. False, cosider your textbook which is a rectagular prism with a o-square rectagular base. 4. A quadrilateral may have all acute agles. False, the sum of the agels is 80(4 ) = 60. If a + b + c + d = 60 ad each oe is < 90, they ca t add up to A quadrilateral may have both a right agle ad a obtuse agle. True

9 44. A semi-regular tessellatio ca be made usig oly regular hexagos ad squares. False, this is b from sectio Every prism has rectagular lateral faces. False, if the prism is oblique, the faces are o-rectagular parallelograms 46. Some scalee triagles are right triagles. True, for example a -4-5 right triagle. 47. Every pyramid has isosceles triagles sides. False, if the pyramid is oblique, the faces are scalee triagles. 48. All quadrilaterals have at least two lies of symmetry. False, see # The top of a coe is called a vertex or apex. True by defiitio \ 50. Ay two regular polygos that tessellate by themselves ca be used together to create a semi-regular tessellatio. Square / triagle Square / hexago Hexago / triagle Does t work. You ca t fid a combiatio of 90 ad 0 that adds up to 60 There are other arragemets too So the aswer is FALSE 5. If a polygo is cocave, the it must have a reflex agle. True, i fact this is a alterate defiitio for cocave. Draw ay cocave polygo to see for yourself. 5. All regular polygos are covex. True, as the umber of sides icrease, the polygos start lookig more ad more circular quarts = 8 ouces 4cups 8ouces True 4 qts 448 ouces 8 ouces qt cup 54. If the area of a circle doubles, so does the radius. False, if the area doubles, the scale factor is so the radius, which is a legth, goes up by the scale factor.

10 55. If the surface area of two cyliders is the same, the the cyliders have the same volume. False, roll a piece of paper the short way ad the log way to see this. 56. A cube of with side legth 0cm would fit iside of a circle with diameter 0cm. Not eve close, the cube would have to pass 5 cm through as this square sectio illustrates ad 5 cm 5 cm the logest side that could fit would be 5 cm 0cm C is warmer tha 50 F. F 00C F 0C True -.8 F.8 Cb.8 (0) b b 00 F.8C F.8(5) 77, 5C is warmer 58. The volume of a right pyramid is times the volume of a right prism with the same base ad height. True Volume pyramid Ah A area of base h height Volume prism Ah A area of base h height 59. The diagoals of a kite itersect at right agles.. True, draw some kites to see this 60. All right triagles are similar. False, for example, there are may isosceles right triagles ad these are ot similar to a -4-5 right triagle 6. Triagle has side legths cm, cm ad 5 cm. Triagle has side legths cm, cm ad 5 cm. Triagle ad Triagle must be similar. True, they are cogruet by SSS so there are also similar. 6. If a rectagular prism is scaled by a factor of, the surface area icreases by a factor of 6. False, icreases by x = 9 6. Triagle has agles 40, 40 ad 00 ad Triagle has agles 40, 00 ad 40. Triagle ad Triagle must be cogruet. True by AA similarity