Polygon Analogy Solution: Octagon and Heptagon Sides

Let's break down the relationship between the words in this analogy question. The question asks us to find the word that relates to 'Heptagon' in the same way that 'Eight' relates to 'Octagon'.

Understanding the First Pair: Octagon and Eight

• The first word is 'Octagon'.
• The second word is 'Eight'.
• An 'Octagon' is a polygon.
• The word 'Octagon' comes from Greek words 'okta' meaning eight and 'gonos' meaning angle.
• Therefore, an Octagon is defined as a polygon with eight sides and eight angles.
• The relationship here is that 'Eight' is the number of sides (or angles) that an 'Octagon' has.

Applying the Logic to the Second Pair: Heptagon and ?

• The third word is 'Heptagon'.
• We need to find the fourth word that relates to 'Heptagon' based on the same rule.
• A 'Heptagon' is also a polygon.
• The word 'Heptagon' comes from Greek words 'hepta' meaning seven and 'gonos' meaning angle.
• Therefore, a Heptagon is defined as a polygon with seven sides and seven angles.
• Following the pattern, the word related to 'Heptagon' should be the number of sides it has.

Identifying the Missing Word

Since a Heptagon has seven sides, the missing word in the analogy "Octagon : Eight :: Heptagon : ?" is Seven.

Checking the Options

Let's look at the given options:

1. Ten
2. Six
3. Nine
4. Seven

The word 'Seven' matches option 4.

Conclusion

The analogy is based on the definition of the polygons. An Octagon has eight sides, and a Heptagon has seven sides. Therefore, the correct relationship is Heptagon : Seven.

Revision Table: Polygon Names and Sides

Understanding the prefixes for polygon names is key to solving this type of analogy. Here is a quick reference:

• 'Tri-' means three
• 'Quadri-' or 'Tetra-' means four
• 'Penta-' means five
• 'Hexa-' means six
• 'Hepta-' or 'Septa-' means seven
• 'Octa-' means eight
• 'Nona-' or 'Enea-' means nine
• 'Deca-' means ten

Additional Information: Regular Polygons

The terms Octagon and Heptagon usually refer to any polygon with that number of sides. However, they are often encountered in the context of "regular" polygons. A regular polygon has all sides equal in length and all interior angles equal in measure.

For example:

• A regular octagon has eight equal sides and eight equal interior angles (each measuring $135^\circ$).
• A regular heptagon has seven equal sides and seven equal interior angles (each measuring approximately $128.57^\circ$).

While the concept of 'regular' wasn't needed to solve this specific analogy (which only focused on the number of sides), it is an important related concept in geometry when studying polygons.