# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## 5. INSIDE OR OUTSIDE

Figure 30 is also a simple closed curve, but now it is not at all easy to see if point A is inside or outside.

A Frenchman named Jordan discovered an easy way of being sure. Just draw a straight line from the point to the outside curve. If it crosses the curve an odd number of times then the point is INSIDE. If the line crosses the curve an even number of times then the point is OUTSIDE.

### INVESTIGATION 5

SUGGESTION: Ask students to draw several different SIMPLE CLOSED CURVES. Choose a point inside each curve and draw a straight line from the chosen point to the outside of the curve. Count how many times the line cuts the curve. Can you now discover a rule?

- a) Figure 31 shows a simple closed curve and two points A and B. Which point is outside the curve? Can you reach B from A without crossing the curve?
- b) Figure 32 also shows a simple closed curve and four points A, B, C, D. Which of the four points are inside the curve? How can you tell?
- c) Draw other simple closed curves. Try to find a rule for deciding which points are inside your curves.