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Thales' Intercept Theorem

Autor:Simona Riva
Thales' Intercept Theorem, concerning a family of parallel lines cut by two transversal lines, states that the lengths of the segments intercepted by two parallel lines on the first transversal are directly proportional to the lengths of the segments intercepted by the same pair of parallels on the other transversal. Formally: Hypothesis: Thesis: . The same proportion holds for any pair of corresponding segments intercepted on the two transversal lines. Explore the Intercept Theorem in the app below, then complete the activities that follow.

Practice Zone

In the app above, use the buttons to display only the lengths of the first pair of segments, and . Move the parallel lines and the transversal lines so that the ratio of the lengths of and is equal to 1. What can you say about the lengths of the two segments? How would you describe the relative position of the two transversal lines?

Now display the lengths of all the segment pairs. Move the parallel lines and the transversal lines so that the ratio of the lengths of and is equal to 1. What can you say about the lengths of all the displayed segment pairs?

Refer to the graph in the app above and choose which of the following proportions is correct.

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Antwort überprüfen (3)

Can we say that, according to Thales' Intercept Theorem, also the proportion holds? Support your answer with a clear explanation of your reasoning.

On the sides and of the parallelogram consider two congruent segments and . Join with and with , and let and respectively be the points of intersection of and with the diagonal of the parallelogram. Prove that .

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