Congruent triangle
similar triangles
1 The corresponding edges are equal and the corresponding edges are proportional.
2 corresponding angles are equal, corresponding angles are equal.
3. The corresponding median lines are equal, and the ratio of the corresponding median lines is equal to the similarity ratio.
The corresponding angular bisectors are equal, and the ratio of the corresponding angular bisectors is equal to the similarity ratio.
5 equals height, and height ratio equals similarity ratio.
6 girth is equal, girth ratio is equal to similarity ratio.
The area ratio of equal area is equal to the square of similarity ratio.
2. Learning this point should pay attention to the problem:
The properties of (1) similar triangles can be obtained by analogy with some properties of congruent triangles.
(2) The area ratio of similar triangles is equal to the square of the similarity ratio. It is necessary to clarify their two relationships: area ratio = (similarity ratio) 2;
The judgment of two similar triangles.
Similar triangles's knowledge is closely related to the circle, so we must learn this part of knowledge well to lay a good foundation for learning this part of knowledge.
This lecture focuses on two issues: first, the proof of proportional formula and equal product formula; Second, the proof and calculation under the condition of double verticality.
I. Proof of equal product formula and proportional formula:
The proof of equal product formula and proportional formula is a common problem in the chapter of similarity. Because such problems change greatly, students often find it difficult. But if we master the basic laws of solving this kind of problems, we can find a solution.
(1) When you encounter an equal product formula (or a proportional formula), first see if you can find a similar triangle.
Equal product formula can be rewritten into proportional formula according to the basic properties of proportion. If there are three non-repeating letters on each side of the proportional formula, you can find a similar triangle.
(2) If the triangle cannot be found in the verified equal product formula or proportional formula or the found triangle is not similar, equal line segment replacement or equal ratio replacement is required. Sometimes it is necessary to add appropriate auxiliary lines to construct parallel lines or similar triangles.
Second, the calculation and proof under the condition of double vertical:
"Double verticality" means: "in Rt△ABC ∠BCA=900, CD⊥AB in D" (as shown in the figure). In this case, the following conclusions are drawn:
( 1)△ADC∽△CDB∽△ACB
(2) Cd2 = ad BD comes from △ADC∽△CDB.
(3) ac2 = ad ab is obtained from △ADC∽△ACB.
(4) BC2 = BD AB comes from △CDB∽△ACB.
(5) AC BC = AB CD comes from the area.
(6) Pythagorean theorem
There are some problems here