Повторення: властивості подібних трикутників

Similar triangles are triangles that are the same shape but do not necessarily have the same area.

These triangles can be identified using several methods: AA (angle angle), SAS (side angle side), SSS (side side side), and HL (hypotenuse leg).

Angle-Angle Similarity

This form of similarity is just as it sounds. If two corresponding and adjacent angles in both triangles are similar, then the triangles are similar to one another, and vice versa.

For example, in this diagram, we see two triangles where where two adjacent angles are congruent to each other. Thus, these two triangles are similar.

Side-Angle-Side Similarity

This form of similarity is also just as it sounds. If the ratio of one side on one triangle to another side on another triangle is the same as the ratio of another side on the first triangle to another side on the other triangle and if the angles between those proportionate sides are equivalent, then those triangles are similar.

For example, in this diagram, AB/YX = AC/ZX = 3/2. Furthermore, $\angle$BAC = $\angle$ZXY. Therefore $\Delta$BAC $\sim$ $\Delta$YXZ.

Side-Side-Side Similarity

In this form of similarity, if the there is a common ratio between two corresponding sides of both triangles, then they are similar.

For example, in this diagram, AB/XY = BC/YZ = AC/XZ = 4/5. Thus, $\Delta$ABC $\sim$ $\Delta$XYZ.

Hypotenuse-Leg Similarity

This form of similarity is exlusively for right triangles. If the ratio of the hypotenuse of both triangles is equivalent to the ratio of one leg of both triangles, then the triangles are similar.

Note: This is essentially side-side-side similarity because if the hypotenuse and one leg has equivalent ratios, then the final side is undoubtedly in that same ratio.

Theorems and Tricks

Side-Splitter Theorem

Given $\Delta$ADE, and a parallel line BC drawn parallel to DE, then AB/BD = AC/CE. Furthermore, we can state that $\Delta$ADE $\sim$ $\Delta$ABC. You can find the diagram here.

Relationship between Area and Similarity

Given that $\Delta$ABC $\sim$ $\Delta$DEF in a side ratio of 1 : $r$, given some ambiguous real number $r$, the ratio of the area of $\Delta$ABC to the area of $\Delta$DEF is 1 : $r^2$.

This is another way of saying Corresponding Parts of Similar Triangles are in Proportion, or CPSTP. That is, if $\triangle ABC \sim \triangle DEF$, then $\angle A \cong \angle D$ but not necessarily $\angle A \cong \angle E$.