1
Determine the total count of triangles visible within the provided diagram.
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Solution: Step 1: Identify and count the simplest triangles (composed of one component each). There are 8 such triangles: GLK, DLJ, DJM, HMN, QRE, IRA, IPA, and FPO.
Step 2: Identify and count triangles composed of two components each. There are 10 such triangles: BDO, CDQ, DLM, PRA, KFI, NEI, HJI, GJI, DKI, and DNI.
Step 3: Identify and count triangles composed of four components each. There are 5 such triangles: DIE, DFI, DOA, DQA, and GHI.
Step 4: Identify and count triangles composed of six components each. There are 2 such triangles: DCA and DBA.
Step 5: Identify and count triangles composed of eight components. There is 1 such triangle: DEF.
Step 6: Identify and count triangles composed of twelve components. There is 1 such triangle: ABC.
Step 7: Sum all identified triangles: 8 + 10 + 5 + 2 + 1 + 1 = 27.
4
From the set of figures (1), (2), (3), (4), and (5), identify the one that is dissimilar.
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Solution: Step 1: Observe the central pentagon and the 'cup' like shapes surrounding it in each figure.
Step 2: For each figure, count how many of these 'cup' shapes are oriented to open towards the central pentagon, and how many open outwards.
Step 3: In figures (1), (3), (4), and (5), a consistent pattern is observed: three cups open towards the pentagon, and two cups open outwards.
Step 4: In figure (2), this pattern is broken; the number of inward and outward-opening cups is different from the consistent pattern observed in the other figures.
Step 5: Therefore, figure (2) is the one that is different from the rest.
7
Determine the total number of parallelograms present in the given figure.
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Solution: Step 1: Count the simplest parallelograms (composed of one component each). There are 6: ABFE, BCGF, CDHG, EFJI, FGKJ, and GHLK.
Step 2: Count parallelograms composed of two components each. There are 7: ACGE, BDHF, EGKI, FHLJ, ABJI, BCKJ, and CDLK.
Step 3: Count parallelograms composed of three components each. There are 2: ADHE and EHLI.
Step 4: Count parallelograms composed of four components each. There are 2: ACKI and BDLJ.
Step 5: Count parallelograms composed of six components. There is 1: ADLI.
Step 6: Sum all identified parallelograms: 6 + 7 + 2 + 2 + 1 = 18.
11
From the provided figures (1), (2), (3), (4), and (5), select the one that is dissimilar to the others.
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Solution: Step 1: Observe the primary geometric element, which appears to be a triangle, within each figure.
Step 2: Pay attention to the orientation or inclination of this triangle in relation to the overall figure or any implied reference axis.
Step 3: Notice that in figures (1), (2), (3), and (5), the triangle's inclination or position follows a consistent pattern or can be rotated to match each other.
Step 4: Identify that figure (4) has the triangle oriented in a different way compared to the others, making its inclination unique.
Step 5: Therefore, figure (4) is the one that is different from the rest.
15
Determine the missing value at the top of a pyramid given its base and pattern rules.
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Solution: Step 1: Analyze the given pyramid structure and identify the pattern.
Step 2: The bottom row of the pyramid is: 3, 1, 4, 2, 5
Step 3: Derive the second row by adding adjacent numbers: 4, 5, 6, 7
Step 4: Derive the third row: 9, 11, 13
Step 5: Derive the fourth row: 20, 24
Step 6: The top number is the sum of the two numbers directly below it: 20 + 28 is not derived from previous steps, re-evaluate the third row to get 10, 12, 14 and then fourth row 24, 28.
Step 7: The top number is 24 + 28 = 52.