# Circles and Distances

All points on the periphery of a circle have the same distance to the midpoint of the circle. This is a fact that is useful when making constructions. To get some feeling for the circle patterns, do the exercises Geometry - Dynamic Patterns.

Suppose that you want to visualize a ladder against a wall, a ladder sliding down along the wall. If you let the x-axis represent the ground and the y-axis the wall, you can place a point A on the wall and a point B on the ground. Then let the ladder have the endpoints A and B.

The problem with this construction is that the length of the ladder will not remain the same when moving the points. If you want to use a constant length, it is easiest to use a circle with constant radius.

### Exercise 1 - Construct a ladder against a wall

• Make sure that the axes are shown.
• Place a point A on the y-axis (the wall).
• Make a circle with A as centre by using the tool Circle with Centre and Radius . Enter a suitable radius. The radius should be a constant number, since the ladder should have constant length.
• Make the intersection point between the circle and the the x-axis and make a ladder between the two points by inserting a segment between the points.

By putting a trace on some points (or other objects) in a construction, new geometrical patterns can appear.

### Exercise 2 - Cat on a ladder using trace

Place a point, the cat, on the middle of the ladder by using the tool Midpoint or Centre .

Right-click on the point representing the cat and choose `Trace On`. Drag the point A on the wall to let the ladder slide along the wall. You can erase a trace by zooming in or out, use the mouse wheel in order to zoom.

Also try to put a trace on the segment representing the ladder.

### Exercise 3 - Cat on a ladder using locus

The trace that is drawn shows the trajectory of the cat as the point A moves along its trajectory. The trajectory of the cat depends on the trajectory of the point A. You can show this trajectory in another way than using the trace. Find the tool Locus , then click on the cat followed by the point A; the trajectory of the cat will be shown.

### Exercise 4 - More cats

Input more points on the ladder. Show the traces or use Locus to show the trajectories of these points. If you want a more realistic ladder, place the point A on a ray along the positive y-axis, in that case the ladder can never fall below the x-axis.

If you want to generate many traces that appear when dragging an object, you can use the spreadsheet. A description of how to do this is found at GeoGebra Tutorial - Spreadsheet.

Save the construction for the exercise on the next page.