8 - Intercept Theorem

Example

I put my hand one metre from me. With the height of my hand (5 cm), I can hide a 400 metre-high mountain standing far away. How far am I from the mountain?

If the straight lines (AD) and (BC) are parallel (it is the case here because both of them are vertical lines), then the numbers are equal. It is the Intercept theorem. The number is not relevant for us here. On the contrary the equality is interesting because it gives and then with a cross product, we can find , so , so .

Intercept Theorem

 If the straight lines (FM) and (CS) are parallel then .

problem : Calculate CN in the following case knowing that (FA) and (CN) are parallel.

The straight lines (FA) et (CN) are parallel, hence according to the intercept theorem, you have :

The reciprocal of the intercept theorem

The reciprocal of the intercept theorem helps to prove that two straight lines are parallel if you know at least 4 lengths.

To prove that the straight lines (XY) et (WZ) are parallel, you calculate separately the ratios then you show that at least two of them are equal :

so according to the reciprocal of the intercept theorem, the lines (XY) and (WZ) are parallel.

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Intercept theorem

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