The intercept theorem and similar triangles
Slide a line parallel to one side of a triangle and see that the ratio it cuts on the other two sides always stays equal. Then try the converse, similar triangles and scale factors.
- UK: Year 9
- France: grade 8
- Germany: grade 9
- Iran: grade 9
- Afghanistan: grade 8
- Syria: grade 8
- Palestine: grade 8
- Ukraine: grade 8
- Russia: grade 8
Guide
For Year 9 (KS3: similarity, enlargement and scale factors) and Iran Grade 9 / Afghan Grade 8. Iranian and Afghan books call it Thales' theorem; in England "Thales' theorem" means the angle in a semicircle, so here it is the intercept theorem. The coloured dots can be dragged and the ratios under the figure update live.
The intercept theorem: if a line parallel to side BC cuts the other two sides at D and E, then AD/DB = AE/EC. The converse is true too: if those two ratios are equal, DE is parallel to BC.
Then triangle ADE is similar to triangle ABC: the angles are equal and all three sides are enlarged by the same factor, AD/AB = AE/AC = DE/BC. That number is the scale factor.
Two triangles with two pairs of equal angles are similar (AA); you don't need to measure the sides. In similar shapes lengths are multiplied by a number, never increased by adding one. On the board ratios are written with ÷.