CliffsNotes Geometry Common Core Quick Review by M. Sunil R. Koswatta

Author:M. Sunil R. Koswatta
Language: eng
Format: epub
Published: 2017-04-14T04:00:00+00:00

Geometry Theorems

The geometry theorems in this book are numbered for organizational purposes. For example, Theorem 2.2 is the second theorem given in Chapter 2. Study these theorems by their content, not by their number, as the theorem numbering has no significance outside of this book.

Length of a Line Segment

Suppose we want to find the length of a given segment . We need a ruler for this. Consider a number line (a ruler) with tick marks at whole numbers, and the first tick mark is at 0 (zero). The length unit of this ruler is the length of the segment [0, 1], that is, 1. For example, if we place a segment on this number line so that one end point is at 0 and if the other end falls on the tick mark 5, then we say the length of the segment is 5 units.

Place the segment on the number line so that A is at 0. Then point B falls on some point on the number line. The point B may fall on a tick mark, or not. If it does fall on a tick mark, let’s say 4, then we say the length of is 4.

If B does not fall on a tick mark, then it lies between two tick marks. For example, suppose B lies between 4 and 5. Then the length of is 4 + t1, for some number t1, where 0 < t1 < 1. Our goal now is to find t1. We adopt the following method.

Partition the segment [4, 5] into 10 equal parts. Then each part has a length of units, or 0.1 units. Then there are nine new tick marks between 4 and 5, and we identify them as 4.1, 4.2, 4.3, 4.4, …, 4.9. If B falls on one of these new tick marks, say 4.4, then we say the length of is 4.4. However, it is possible that B falls between two of the new tick marks and we still do not know the length of . Suppose B lies between 4.4 and 4.5. Then the length of is 4.4 + t2, for some number t2, where 0 < t2 < 0.1. Our goal now is to find t2. We adopt the same method as before.

Partition the segment [4.4, 4.5] into 10 equal parts. Then each part has a length of units, or 0.01 units. Then there are nine new tick marks between 4.4 and 4.5, and we identify them as 4.41, 4.42, 4.43, …, 4.49.

If B falls on one of these new tick marks, then great. Suppose B is on the tick mark 4.48. Then we say that the length of is 4.48. However, it is possible that B lies between two of the new tick marks. Then we still do not know the length of .

Suppose B lies between 4.48 and 4.49. Then the length of is 4.48 + t3, for some number t3, where 0 < t3 < 0.01.

Partition the segment [4.48, 4.49] into 10 equal parts, obtaining nine new tick marks between 4.

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