lameda USD Geometr Benchmark Stud Guide ind the perimeter and area of the rhombus. ind the perimeter and area of the trapezoid. 0 0 0 erimeter o find the perimeter, add up the lengths of all the sides. ind the length of each side using thagorean heorem: let side length + ± erimeter un + un + un + un un 0 un herefore the length of each side is units rea i) Using a formula for the area of a rhombus where d and d are the lengths of the diagonals rhombus rhombus rhombus rhombus dd un un ( un ) un ii) Decomposing the area of the rhombus into right s: bh rhombus ( ) rhombus ( un ) ( un )( un ) rhombus un ( un ) un G.MG. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide ind the perimeter and area of the regular heagon. Determine whether the following statements are true or false given the regular octagon below: B in. erimeter o find the perimeter, add up the lengths of all the sides. Since the heagon is regular all of the sides would have a length of in. erimeter in + in + in + in + in + in in in H G C D rea 0 ft. E in. 0 measure of a central angle n 0 0. BE is a rectangle. B. BCDE is an isosceles trapezoid. 0 a a 0 C. 0 ft in. i)using a formula to find the area of a regular polgon where a is the length of the apothem and is the perimeter: regular polgon heagon heagon a in in 9 in 0 or a 0 0 90 we can find the long leg b multipling the short leg b : a ( ) a ii) Decomposing the area of the heagon into si s: bh ( in )( in ) heagon heagon heagon in ( ) in 9 in G.MG. D. he perimeter of the octagon is 0 ft. E. he area of the octagon is ( 00 + 00 ) ft age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide ind the surface area and volume of the triangular prism. Determine whether the following statements are true or false given the right triangular prism. 0 ft. 0 ft. Surface rea 0 ft. ft 0 0 ft. 0 ft. h 0 0 ft. 0 0 0 ft. bh 0 ft ft ft 0 ft or a 0 0 90 we can find the long leg b multipling the short leg b : h h cm cm cm. he prism has faces. B. he prism has 9 edges. C. he area of the base is cm. D. he surface area of the prism is E. he volume of the prism is cm. 0 cm. Using a ormula S.. B + h prism B + h + 0 0 + 0 ( 0 0 ) ft + Using a Net 0 0 0 0 0 ( ) ( rectangle ) S.. + + 0 0 + 0 ( 0 0 ) ft + Volume V Bh prism V B h prism V ft ft prism V 00 ft prism G.MG., G.GMD. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide ind the surface area and volume of the clinder. Determine whether the following statements are true or false given the right clinder. in ft O B Given: B and CD are diameters, B CD, O is the midpoint of B, O ft a) ind the area and circumference of the base. he base is a with a diameter of ft:. ft. ft ft (. ft) π r π π ft π ft π ft b) ind the surface area of the cone. S.. B + h prism S.. B + Ch clinder S π S.. clinder π ft + 0 π ft 0 S.. clinder π ft + π ft S.. clinder π ft ( π ).. clinder ft + ft ft c) ind the volume of the cone. V prism clinder i π ft ii π ft 00 π ft clinder π ft clinder clinder clinder Bh V B h V V V V ( ft) in C dπ C ( ft) C π ft clinder is similar to a prism with a circular base clinder is similar to a prism with a circular base π D. BCD is a square. B. C ft C C. DO ft D. he surface area of the clinder is E. he volume of the prism is π ft 9 π ft G.MG., G.GMD., G.GMD. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide he circular base of the right cone below has a diameter of ft. Given the square pramid below: 0 ft ft d a) ind the area and circumference of the base. he base is a with a diameter of ft: 0 d a) ind the area of a lateral face of the square pramid. ft ft ft π r π π ( ft) ( ft ) π ft C dπ C ( ft) C π ft π b) ind the surface area of the cone. S.. pramid B + l S.. cone B + Cl S.. cone ft + ft 0 ft S ( π ) ( π ) ( π ).. cone π ft + ft 0 ft S.. π ft + 0 π ft cone S.. π ft cone cone is similar to a pramid with a circular base b) ind the surface area of the square pramid. c) ind the volume of the cone. 0 ft 0 ft h -- right multiplied ft b a scale factor of would produce a --0 right. herefore, h ft. Vpramid Bh cone is similar to a pramid with a circular base Vcone Bh V ( ft )( ft cone π ) V 7 π ft cone ft c) ind the volume of the square pramid. G.MG., G.GMD. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide ind the surface area and volume of the composite solid below consisting of a pramid on top of a cube. ind the surface area and volume of the composite solid below consisting of a hemisphere, clinder and cone. 0 in 0 in cm cm cm 0 in 0 in Surface rea he surface area of the solid consists of all the faces of the pramid ecept for the base( s), and all the faces of the cube ecept for the top ( squares). 0 0 0 in 0 l a + b c + 0 l l l l Using ormulas S.. B + l B + ( B + h B) S.. l + B + h S.. ( 0)( ) + ( 00) + ( 0)( 0) S.. 00 + 00 inding the rea of Each ace ( ) ( square ) S.. + S.. bh + ( s ) S.. 0 + 0 S.. + 00 S.. 00 + 00 in S.. 00 + 00 S.. 00 + 00 in Volume o find the volume of the composite solid add the volumes of the pramid and cube. V V + V solid pramid cube Vsolid Bh + ( Bh) Vsolid 00 0 + 00 0 () V V solid solid 000 + 000 000 000 + V V solid solid 000 in in G.MG., G.GMD. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide 7 Determine the shape of each cross section. a) arallel to the base 7 Name all possible cross section shapes of the following solids. a) Sphere Circle b) Or from above b) Square ramid Rectangle c) Or from above rapezoid G.GMD. age 7 of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide he two pramids below are similar with corresponding side lengths of cm and cm. igure igure he two figures below are similar. he ratio of two corresponding side lengths from igure to igure is :. Use this information to determine whether each statement is rue or alse. igure cm cm a) ind the ratio of corresponding sides, surface areas, and volumes from igure to igure. he ratio of corresponding sides ( a : b ) : he ratio of surface areas ( a : b ) : : he ratio of volumes ( a : b ) : : b) If the surface area of igure is surface area of igure. cm, find the Use the ratio of their surface areas to set up a proportion: If the surface area of igure is i cm, then the surface area of igure is 00 cm 00 00 c) If the volume of igure is 000 cm, find the volume of igure. Use the ratio of their volumes to set up a proportion: 000 i000 i If the volume of igure is 000 cm, then the volume of 000 igure is cm G.GMD. igure. If igure has a side length of cm, then the corresponding side length in igure is cm. B. If igure has a surface area of 0 ft, then the surface area of igure is 0 ft. C. If igure has a volume of m, then the volume of igure is m. D. If the area of a face in igure is 7 in, then the area of its corresponding face in igure is in. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide 9 Given: - 9 oint is transformed to form the image. - a) Determine the coordinates of if point was rotated 90 about the origin. **Rotations are counter clockwise (unless stated otherwise)** - - - - - - (, ) b) Determine the coordinates of if point was reflected over the line.. If is rotated 0 about the origin then the coordinates of,. are - (, ) - - - B. If is rotated 90 clockwise about the origin then the,. coordinates of are C. If is translated units to the left and unit down then the,. coordinates of are c) Determine the coordinates of if point was reflected over the line. D. If is reflected across the - ais, then the coordinates of are (, ). - - (, ) - - E. If is reflected across the line, then the coordinates of are (, ). G.CO. age 9 of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide 0 Draw the resulting image after each of the following transformations takes place on this preimage: 0 Draw the resulting image after each of the following transformations takes place on this preimage: - B C E - a) 90 clockwise rotation about the origin. - G B H B - C - - - (a,b), ( ) B, C, 90 clockwise rotation about the origin - C (b,-a), ( ) B, C, a) 90 rotation about the origin. - E G b) Dilation of centered at the origin. H - - B C - b) Dilation of centered at the origin. - - E (a,b), ( ) B, C, Dilation of centered at the origin a, b ( ) ( ), B, C, - H - G G.CO., G.SR. age 0 of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr wo si-sided dice are rolled. a) How man outcomes are possible? outcomes - - - - - - b) What is the probabilit that the sum of the two dice is eactl? ossibilities - - - - - - - - - - - ( sum is ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - c) What is the probabilit that the sum of both dice is greater than or equal to? inding all possible sums greater than or equal to : - - - - - - - - - - - - Benchmark Stud Guide random integer is selected from to 0. a) What is the probabilit that the number is a multiple of? b) What is the probabilit that the number is a perfect square? - - - - - - - - - - - - - - - - - - - - - - - - 0 ( sum ) ( sum ) - - - - - - - - - - - - c) What is the probabilit that a double digit number is selected? Using the complement: he sum of the dice will either be less than or greater than or equal to 00% of the time: ( ) + ( sum < ) ( sum ) ( sum < ) sum ( ) sum ( ) sum ( ) sum ( u ) s m 0 ossibilities of a sum < (the und outcomes above) - - - - - - S.C. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide wo si-sided dice are rolled. a) ind the probabilit that the second die is even given that the first die is odd. Would the probabilit be different if the first die was even? Wh or wh not? - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - random integer is selected from to 0. a) ind the probabilit that the number is a multiple of given that it is even. Would the probabilit be different if it was given that it was odd? Wh or wh not? 9 (even given st die is odd) he probabilit would not be different if the first die was even since these are independent events. he first roll does not affect the likelihood of rolling an even on the second roll. b) ind the probabilit that the sum is greater than 7 given that the first die is a. Would the probabilit be different if the first die is a? Wh or wh not? - - - - - - (sum is greater than 7 given st die is a ) he probabilit would be different since these are not independent events. he first roll does affect the sum and if it was a, then the probabilit that the sum is greater than 7 would be. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - b) ind the probabilit that the number is a perfect square given that it is a multiple of. Would the probabilit that the number is a multiple of given that it is a perfect square differ? Wh or wh not? S.C.. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide random integer is selected from to 0. coin is flipped and a ten-sided die is rolled. a) What is the probabilit that the number is even and a multiple of? 7 9 0 7 9 0 7 9 0 We can look at all the possible outcomes and see which ones are even (underlined) and a multiple of (d):,,,, 0 (mult. of and even) 0 Since the two events are independent we can multipl their individual probabilities: (even and mult. of ) (even) i(mult. of ) 0 i 0 0 0 i 0 0 b) What is the probabilit that the number is even or a multiple of? 7 9 0 7 9 0 7 9 0 We can look at all the possible outcomes and see which ones are even (underlined) or a multiple of (d):,,,, 0,,,,, 0,,,,, 0, he two events are not mutuall eclusive since there are overlapping events where the numbers are even and a multiple of. We can find the probabilit b adding the individual probabilities and subtracting the outcomes that overlap:. Getting heads and rolling a 9 are mutuall eclusive events. B. he probabilit of getting heads and rolling a 9 is 0. C. he probabilit of getting heads or rolling a 9 is. D. he probabilit of getting tails and rolling an even number is. E. he probabilit of getting tails or rolling an even number is.,, 9,,,,,, 7, 0 alread accounted for 0 (even or mult. of ) 0 (E or Mof) (E) + (Mof) (E and Mof) 0 + 0 0 0 0 0 0 0 ** Note: Events and B are mutuall eclusive if there is no overlap between the events: ( and B)0 S.C.7, S.C. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr lonzo, Beatrice, Chad, Delmon, and Eunice are the onl contestants in a 00 d race. st, nd, and rd place qualif for the State inals. a) How man possible orders are possible for st, nd, and rd place? Using,B,C,D, and E for each of the runners BC BC CB DB EB BD BD CD DC EC BE BE CE DE ED CB BC CB DB EB CD BCD CBD DBC EBC CE BCE CBE DBE EBD DB BD CD DC EC DC BDC CDB DCB ECB DE BDE CDE DCE ECD EB BE CE DE ED EC BEC CEB DEB EDB ED BED CED DEC EDC b) What is the probabilit that Chad finishes st and lonzo finishes nd? here are permutations that meet the criteria: CB, CD, CE With Chad st and lonzo nd, there are remaining letters to go rd : osibilities with C st and nd i i (Chad and lonzo ) 0 0 st nd c) What is the probabilit that Beatrice, Delmon, and Eunice qualif (all finish in the top )? B C D E C D E B D E B C E B C D here are letters that can be st. or each letter that goes first, there are remaining letters to go nd. or each of those letter permutations (0) there are remaining letters to go rd. ossible orders i i 0 Benchmark Stud Guide club sponsor is choosing her new president and vice president from the si returning members: Jordan, Kree, Ldia, Mirna, Nahim, and Omar. She decides to put the names in a hat and randoml select the positions. a) How man results are possible? b) What is the probabilit that Omar is chosen to be president? c) If the club sponsor decides to have co-presidents instead of a president and vice president before she draws the names, what is the probabilit that Jordan and Ldia are co-presidents together? Using ermutations (order matters) BDE, BED, DBE, osibilities with B,D,E in the top i i DEB, EBD, EDB (B,D,E in the top ) 0 0 Using Combinations (order doesn t matter) he permutations BDE, BED, DBE, DEB, EBD, and EDB equate to onl one combination of the letters B,D, and E. he total possible combinations of qualifiers would be 0:,B,C,B,D,B,E,C,D,C,E,C,E B,C,D B,C,E B,D,E C,D,E! C!! 0 (B,D,E in the top ) 0 S.C.9 End of Stud Guide age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr You r Solutions: Benchmark Stud Guide ind the perimeter and area of the trapezoid. erimeter erimeter 0 un + 0 un + 0 un + un un rea o find the area we are going to need the height 0. BE is a rectangle. B. BCDE is an isosceles trapezoid. C. 0 ft 0 0 D. he perimeter of the octagon is 0 ft. We can find the height using the thagorean heorem with one of the right s: a + b c + h ( 0) + h 00 h h h ± h un i) We can use a formula for the area of a trapezoid where b and b are the lengths of the bases trapezoid ( b + b ) h trapezoid ( 0 un + un )( un ) trapezoid ( un )( un ) ( un)( un) 0 trapezoid trapezoid un E. he area of the octagon is ( 00 + 00 ) ft. he prism has faces. B. he prism has 9 edges. C. he area of the base is cm. D. he surface area of the prism is E. he volume of the prism is 7 cm. 0 cm. ii) We can decompose the area of the trapezoid into a rectangle and right s: trapezoid rectangle + bh ( ) trapezoid 0 un un + un ( un )( un ) trapezoid 0 un + un un trapezoid ( un ) un age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr d. BCD is a square. B. C ft C. DO ft D. he surface area of the clinder is E. he volume of the prism is Given the square pramid below: π ft 9 π ft a) ind the area of a lateral face of the square pramid. 0 d bh 0 d d d b) ind the surface area of the square pramid. S.. pramid B + l S.. pramid square + ( ) S.. pramid ( 00 d ) + ( d ) S.. pramid ( 00 d ) + ( 0 d)( d) S.. 00 d + 0 d pramid S.. 0 d pramid c) ind the volume of the square pramid. Vpramid Bh Vpramid ( 00 d )( d ) V 00 d pramid d l d his is a -- right. herefore, l d. S.. 00 d + 0 d pramid S.. 0 d pramid 7 Benchmark Stud Guide Surface rea We can find the surface area b adding the surface areas of the hemisphere, clinder(ecluding the bases) and cone (ecluding the base). S.. ( π r ) ( ) + B + Ch B + B + Cl B S.. π r + Ch + Cl S.. π + ( π ) + ( π ) S.. π + π + π S.. π cm Volume We can find the volume b adding the volumes of the hemisphere, clinder and cone. V π r + π r h + π r h V π + π + π ( ) V π + 7π + π V 0 π cm Name all possible cross section shapes of the following solids. a) Sphere Circle, point b) Square ramid raingle, square, trapezoid, kite, point. If igure has a side length of cm, then the corresponding side length in igure is cm. B. If igure has a surface area of 0 ft, then the surface area of igure is 0 ft. C. If igure has a volume of m, then the volume of igure is m. D. If the area of a face in igure is 7 in, then the area of its corresponding face in igure is in. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide 9. If is rotated 0 about the origin then,. the coordinates of are 0 a) 90 rotation about the origin. B. If is rotated 90 clockwise about the origin then the coordinates of are,. - E G G C. If is translated units to the left and unit down then the coordinates of are,. E H - H D. If is reflected across the -ais, then the,. coordinates of are b) Dilation of centered at the origin. E E. If is reflected across the line, then the coordinates of are (, ). - E G G H H - age 7 of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr a) What is the probabilit that the number is a multiple of? ossibilities 0 ( mult. of ) ( mult. of ) 0 b) What is the probabilit that the number is a perfect square? ossibilities 0 7 9 ( perfect square) ( perfect square) 0 c) What is the probabilit that a double digit number is selected? 0 7 9 7 9 9 7 ( double digit) + ( single digit) ( double digit ) ( single digit ) 0 0 0 ( double digit ) ( double digit ) ( double digit ) 9 7 9 7 7 9 0 9 0 0 0 7 0 0 0 0 7 9 7 9 0 0 Single Digit ossibilities,,,,,, 7,, 9 Benchmark Stud Guide a) ind the probabilit that the number is a multiple of given that it is even. Would the probabilit be different if it was given that it was odd? Wh or wh not? 7 9 (mult. of given even) he probabilit that the number is a multiple of given that it is odd would be different. Given that a number is odd, it would be impossible to be a multiple of, so the probabilit would equal zero. b) ind the probabilit that the number is a perfect square given that it is a multiple of. Would the probabilit that the number is a multiple of given that it is a perfect square differ? Wh or wh not? 7 9 (perfect square given multiple of ) 0 he probabilit that the number is a multiple of given that it is a perfect square would be different: 7 9 0 0 0 9 7 9 7 9 7 0 0 0 7 9 7 9 7 9 (multiple of given perfect square) 0 0 0 Changing what is given affects the amount of possible outcomes. When it is given that the number is a multiple of, there are 0 possible outcomes, of which was a perfect square. When it is given that the number is a perfect square, there are possible outcomes, of which is a multiple of. age of 9 MCC@WCCUSD (USD) 0/7/
lameda USD Geometr Benchmark Stud Guide. Getting heads and rolling a 9 are mutuall eclusive events. B. he probabilit of getting heads and rolling a 9 is 0. a) How man results are possible? Using J,K,L,M, N and O for each of the members i 0 results JK KJ LJ MJ NJ OJ JL KL LK MK NK OK JM KM LM ML NL OL JN KN LN MN NM OM JO KO LO MO NO ON C. he probabilit of getting heads or rolling a 9 is. D. he probabilit of getting tails and rolling an even number is. b) What is the probabilit that Omar is chosen to be president? JK KJ LJ MJ NJ OJ JL KL LK MK NK OK JM KM LM ML NL OL JN KN LN MN NM OM JO KO LO MO NO ON (Omar is resident) 0 E. he probabilit of getting tails or rolling an even number is. c) If the club sponsor decides to have copresidents instead of a president and vice president before she draws the names, what is the probabilit that Jordan and Ldia are copresidents together? Using ermutations (order matters) here are two results that would have Ldia and Jordan as co-presidents(jl and LJ) out of 0 total results: (Ldia and Jordan are co-presidents) 0 Using Combinations (order does not matter) here is one combination where Ldia and Jordan would be co-presidents(j,l) out of possible combinations: J,K J,L J,M J,N J,O K,L K,M K,N K,O L,M L,N L,O M,N M,O N,O (Ldia and Jordan are co-presidents) age 9 of 9 MCC@WCCUSD (USD) 0/7/