Metric relations in the right triangle and properties of a proportion
Draw the altitude to the hypotenuse of a right triangle and see that AH² = BH×HC; find a side with it, use the adding property of a proportion, and build Pythagoras from the same figure.
- Germany: grade 9
- Iran: grade 10
- Russia: grade 8
Guide
This studio follows Iran's Grade 10 Geometry 1 (chapter 2).
In triangle ABC the angle at A is right and AH is the altitude (height of a triangle) to the hypotenuse. The two small triangles ABH and ACH are similar to ABC, so equal ratios give AH² = BH×HC, AB² = BH×BC and AC² = HC×BC.
Property of a proportion: if ab = cd then a + bb = c + dd.
Adding the two leg rules gives AB² + AC² = (BH + HC)×BC = BC²: Pythagoras' theorem.