# Introduction to Geometric Proportionality

This App features a series of games about geometric proportionality. The figure below shows a circle with an arc and the same curves unwrapped into lines. The points are located proportionally along their respective parents (the full line segment and the circle). Points B and G (in green) are located proportionally at t, while points C and F are at -t.

The lengths of the line segment and arc connecting these points are displayed below; notice that they are identical. Drag the slider and press the "Animate" button to see what happens when the value of t changes.

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 t 0 0 6.2831853 BC GF

As you can see, the length of the line segment is always the same as the absolute value of the length of the curve. The negative sign is a result of the orientation of the two points on the circle; notice how the arc switches directions when the points cross at t = pi.

Can you infer the radius of the circle from the lengths of the arc and line segment? Remember that the long white line segment represents the circumference of the circle, and the arc and smaller line segment are the same size. Try adjusting t so that

Let's look at one more figure to see exactly what's happening with this proportionality. This static animation shows a single point located proportionally at t around a circle. The point's position at equally spaced intervals of t is traced out.

As you can see, the values of t are equally spaced around the circle; in other words, our proportional point at t moves around the circle at a constant rate. Through each level of this game, your challenge will be to determine how a point is located proportionaly on a curve as a function of t.

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App generated by Geometry Expressions