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Unit 48 – Geometric Analogies

Difficulty: Very Hard

Time: 3 minutes

The segment below has been adapted from How To Solve It by George Polya.

Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts.

A rectangular parallelogram is analogous to a rectangular parallelepiped. In fact, the relations between the sides of the parallelogram are similar to those between the faces of the parallelepiped: Each side of the parallelogram is parallel to just one other side, and is perpendicular to the remaining sides. Each face of the parallelepiped is parallel to just one other face, and is perpendicular to the remaining faces. Let us agree to call a side a “bounding element” of the parallelogram and a face a “bounding element” of the parallelepiped.

Then, we may contract the two foregoing statements into one that applies equally to both figures: Each bounding element is parallel to just one other bounding element and is perpendicular to the remaining bounding elements.

Thus, we have expressed certain relations which are common to the two systems of objects we compared, sides of the rectangle and faces of the rectangular parallelepiped. The analogy of these systems consists in this community of relations.

1 – In terms of the explanation given above, which of the following pairs is analogous?

  • A. Kettle // Hot Water
  • B. Cigar // Cigarette
  • C. Food // Sandwich
  • D. Hat // Head

2 – Analogy is to similarity as parallelipid is to:

  • A. Rectangle
  • B. Shape
  • C. Bounding element
  • D. Geometry

Highlight the text boxes below to reveal the answers:

Q1: B

Cigars and cigarettes are geometrically analogous to one another; they are both cylinders. Kettles make water hot, a sandwich is a type of food, and a hat goes on your head. They are related conceptually, but in terms of their relative physical parts they don’t necessarily have anything in common.

Q2: B

Line 1: “Analogy is a sort of similarity” – like how a parallelipid is a type of shape. A parallelipid is a 3D shape, whereas a rectangle is 2D, it’s therefore not a type of rectangle – in the same way a cube is not a type of square, but can be thought of as a square extended into 3D space. B is therefore the better answer.