Question 1: In the quadrilateral ABCD, the diagonals AC and BD intersect at X. AX = BX and DX = CX. AB is parallel to DC. a) Name a pair of congruent triangles. b) Name a pair of similar triangles. a) AXD & BXC (SAS) b) A X B & C X D
Question 2: In the diagram, CEFG is a rectangle and ABCD is a square. ABG is a straight line. a) Show that DCE BCG. b) Prove that triangle DCE is congruent to triangle BCG. c) Hence show that CEFG is a square. a) DCB 90 DCE 90 ECB BCG 90 ECG DCE BCG b) DC BC EDC GBC DCE BCG By ASA, DCE is congruent to BCG c) Since DCE is congruent to BCG, EC = GC = FG = EF CEFG is a square
Question 3: In the diagram, AFB, AEC and FED are straight lines. DE = 7 cm, FED//BC, AFB//DC and AE = EC. a) Name a triangle that is congruent to AEF. State the reasons for its congruency. b) Name a triangle that is similar to AEF. State the reasons for its similarity. c) Find the length of BC. a) b) CED (ASA) ACB (AAA) c) AE AC FE BC 1 2 FE BC 1 2 7 BC BC 14cm
Question 4: In the diagram, ADC is a straight line and ABD BCD. a) Prove that triangle ABD is similar to triangle ACB. State your reasons. b) AD = 4 cm, BD = 6 cm and BC = 9 cm. i. Prove that triangle ABD is an isosceles triangle. ii. Find BAD. a) ABD = ACB (given) BAD = CAB (common) BDA CBA AAA, both are similar b) i. AD AB BD CB 6 9 4 AB 6 9 AB 6 AB = BD = 6 ABD is an isosceles ii. Using cosine rule, 6 2 6 2 4 2 2(6)(4)cos cosâ 1 3  70.5 (1dp)
Question 5: In the figure below, rectangle ABCD is reduced to rectangle EFGH. From the information above, find the value of p. 4 p 3 9 5 3 4 p 3 3 5 4 p 3 15 4 p 18 p 4.5cm
Question 6: Given that RP = 9 cm and PQ = 3 cm, a) show that TPQ is similar to SPR. b) calculate area of TPQ area of SPR as a fraction. c) QPT is mapped onto SPR by enlargement. State the scale factor of enlargement. a) TPQ SPR (vertically opposite ) QTP RSP (same segment) TPQ is similar to SPR (By AAA) b) ( 3 9 )2 1 9 c) 3
Question 7: In the diagram below, AB = 8 cm, BC = 5 cm, AE = 10 cm and BE // CD. a) Show that triangle ABE is similar to triangle ACD. b) Find the value of i. BE. CD ii. ED. iii. area of ΔABE. area of ΔACD iv. v. area of ΔABE. area of trapezium BCDE area of ΔBCD. area of ΔABD a) ABE = ACD AED = ADC By AAA, ABE is similar to ACD b) iv. BE CD AB AC 8 13 i. ( 8 13 )2 64 169 v. AE AD = AB AC AD AE = AC AB AD 16 1 4 cm ED = 6.25cm ii. 64 105 iii. BE:CD 8:13 21 parts 105 13 parts 65 65 104 5 8
Question 8: In the diagram below, D and E are two points on ABC in which DE // BC. Given that CD and BE intersect at F. AE = 4 cm and EC = 12 cm. a) Show that triangles ABC and ADE are similar. b) Find the length of DE if BC = 18 cm. c) Find the value of area of ADE. area of quadrilate ral DBCE a) ADE = ABE AED = ACB DAE = BAC By AAA, ABC is similar to ADE b) DE BC = AE AC DE 18 4 16 DE = 4.5 c) ADE ABC ( 1 4 )2 1 16 ADE DBCE 1 15
Question 9: The two containers shown in the diagram are geometrically similar. Their height are 24 cm and 80 cm respectively. a) The diameter of the base of the smaller container is 15 cm. Calculate the diameter of the base of the larger container. b) Find the ratio of the volume of the smaller container to the larger container. Give your answer as a fraction in its lowest term. c) The containers are completely filled with rice. The larger container can hold 250 kg of rice. Calculate the mass of rice that the smaller container can hold. a) 24 80 15 Large Large 50cm b) ( 3 10 )3 27 1000 c) 27 :1000 6.75kg:250kg
Question 10: The two containers shown in the diagram are geometrically similar. Their heights are 20 cm and 50 cm. a) The diameter of the base of the larger container is 14 cm. Calculate the diameter of the base of the smaller container. b) The containers are completely filled with sand. Given that the smaller container holds 2.8 kg of sand, estimate the mass of sand the larger container holds. a) 20 50 smaller 14 5.6 smaller b) ( 2 5 )3 8 125 8:125 2.8:43.75kg
Question 11: Two cups are geometrically similar in shape. The total surface area of the smaller cup is 8 cm 2 and the total surface area of the larger cup is 18 cm 2. a) Find the ratio between the heights of the two cups. b) The smaller cup fruit juice is sold at $0.80 per cup. Adam wishes to buy the larger cup fruit juice. How much will it cost him? a) 8 :18 4 : 9 ht 2 :3 b) V1 V2 ( 2 3 )3 8 27 8 :27 $0.80 : $2.70
Question 12: The base areas of two geometrically similar vases are in the ratio of 16 : 25. a) The curved surface area of the larger vase is 625 cm 2. Find the curved surface area of the smaller vase. b) If the mass of the smaller vase is 4.8 kg, find the mass of the larger vase. a) 16 :25 400 :625 b) ht 1 ht 2 16 25 4 5 V1 V2 4.8kg x ( 4 5 )3 4.8 x x 9.375kg
Question 13: The diagram below shows a cup in the form of an inverted cone of height 15 cm. Water is poured into the cup until the height of the water is 3 cm. If the volume of the water is 12 cm 3, find the volume of the unfilled space of the cup. ( 3 15 )3 ( 1 5 )3 1 125 1:125 12:?? 1500 Empty space = 1488cm 3
Question 14: The following two thermo flasks are geometrically similar and have capacities of 432 ml and 2 l. a) Find, in its simplest integer form, the ratio of the height of the smaller thermo flask to the height of the larger thermo flask. b) The base area of the smaller thermo flask is 45 cm 2. Find the base area of the larger thermo flask. c) Given the height of the larger thermo flask is 32 cm, find the height of the smaller thermo flask. a) 432:2000 27:125 ( 27 3 125 ) 3 5 3:5 b) ( 3 5 )2 9 25 9 :25 45 :125 c) 3:5 19.2 :32
Question 15: A map is drawn to a scale of 1 : 50 000. a) An expressway leading to East Coast Park is represented by a line of length 5.6 cm on the map. Calculate the actual length of the expressway, giving your answer in kilometres. b) If the actual area of the East Coast Park is 5 km 2, find the area of East Coast Park on the map in cm 2. a) 1 : 50 000 1: 500m 1: 0.5km 5.6 : 2.8km b) 1 : 0.5 km 1 2 : 0.25km 2 20cm 2 : 5km 2
Question 16: A map is drawn to a scale of 1 cm : 8 km. a) Write down the scale in the form 1 : n. b) On the map, two towns are 3.2 cm apart. What is the actual distance between the two towns, in kilometres? c) A plot of land has an area of 96 km 2. Find the area on the map in square centimetres that represents this actual area. a) 1cm :8000m 1cm :800000cm b) 1cm :8km 3.2cm :25.6km c) 1cm 2 8 2 km 2 1cm 2 :64km 2 1.5cm 2 :96km 2
Question 17: An actual ground distance of 5 km is represented by a distance of 2 cm on a city map. a) Express the scale of the map in the form 1 : n. b) If the distance between two buildings on the map is 15 cm find their actual distance on the ground in km. c) If the area of a stadium is 0.3 cm 2 on the map, find its actual area in km 2. a) 2cm :5km 2cm :5000m 2cm :500000cm 1cm : 250000cm b) 2cm :5km 15cm :37.5km c) 2 2 cm 2 :5 2 km 2 4cm 2 :25km 2 0.3cm 2 :1.875km 2
Question 18: An actual area of a pond of 125 m 2 is represented by 5 cm 2 on a map. Calculate the a) scale of the map in the form 1 : n. b) length of a line on the map, which represents a path 13 m long. a) 5cm 2 :125m 2 1cm 2 :25m 2 1cm:5m 1cm:500cm b) 1cm :5m 2.6cm:13m
Question 19: An actual region of a plot of land 36 km 2 is represented by an area of 4 cm 2 on the map. a) If the area of a town on the map is 0.3 cm 2, find its actual area in km 2. b) Find the scale of the map in the form 1 : n. c) Find the length on the map which represents an actual length of 4.5 km. a) 4cm 2 :36km 2 1cm 2 :9km 2 0.3cm 2 :2.7km 2 b) 1cm :3km 1cm : 3000m 1cm : 300000cm c) 1cm :3km 1.5cm : 4.5km
Question 20: The actual area of a pond which is16 km 2 is represented on map A by an area of 400 cm 2. a) If map A is drawn to a scale of 1 : k, find the value of k. b) The distance of two locations on map A is 8.6 cm. Find the distance of the same two locations on map B which has a scale of 1 cm : 0.4 km. c) State which map has a longer map distance of the two locations. a) 400cm 2 :16km 2 25cm 2 :1km 2 5cm:1km 5cm:1000m 5cm:100000cm 1cm:20000cm b) 1cm :20000cm 8.6cm:172000cm 1cm:0.4km 1cm:400m 1cm:40000cm 4.3cm:172000cm c) 8.6cm, Map A
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