Summary of "중2(하) 도형의 닮음: 개념부터 완벽하게 정리하자!!"

Overview

This lesson (middle-school level) explains geometric similarity (닮음). It defines similarity, shows how to write and read similarity notation, states basic properties for plane and solid figures, gives triangle-similarity criteria (SSS, SAS, AA), and treats the special case of right-triangle similarity (including the three standard relations that follow when you drop the altitude to the hypotenuse). The teacher emphasizes matching corresponding points in the same order, expressing the scale factor in simplest form, and memorizing the right-triangle formulas.

Key concepts and definitions

Definition of similarity: Two figures are similar if one is an enlargement or reduction of the other by a constant ratio (scale factor). Equivalently: corresponding angles are equal and corresponding side lengths are proportional.
Similarity (scale) ratio: the constant of proportionality between corresponding linear measures. Usually expressed in simplest form (e.g., 1:2 rather than 2:4). Often written as k or a:b.
Notation: use the similarity symbol (~) and list vertices in corresponding order. Example: △ABC ~ △DEF means A ↔ D, B ↔ E, C ↔ F. Order must match corresponding points.

Properties of similar figures

For plane figures:
- All corresponding side lengths are in the same ratio (a single constant).
- All corresponding angles are equal.
For solids (3D figures):
- Corresponding edges scale by the linear factor k.
- Corresponding areas scale by k^2.
- Corresponding volumes scale by k^3.
Examples of always-similar families: all circles are similar; all squares are similar; regular polygons with the same number of sides are similar.

How to write and interpret similarity statements

Identify corresponding vertices by matching equal angles or by orientation.
Write triangles in the same vertex order: e.g., △ABC ~ △DEF implies A ↔ D, B ↔ E, C ↔ F.
Do not swap vertex order arbitrarily — corresponding points must line up.

How to compute and use the similarity (scale) ratio

Pick a pair of corresponding sides and form their ratio (e.g., AB : DE).
Reduce the ratio to simplest integer form if asked.
Use the same ratio for any other pair of corresponding linear measures:
- AB/DE = BC/EF = AC/DF = k (the scale factor).
For solids: use k for edges, k^2 for areas, and k^3 for volumes when converting measures.

Triangle similarity criteria

SSS (Side-Side-Side): If three pairs of corresponding sides are in proportion, the triangles are similar.
SAS (Side-Angle-Side): If two pairs of corresponding sides are in proportion and the included angles are equal, the triangles are similar.
AA (Angle-Angle): If two pairs of corresponding angles are equal, the triangles are similar (the third angle then matches automatically).

Using similarity in problems — step-by-step

Identify which shapes or triangles could be similar (look for equal angles or proportional sides).
Determine correspondence (match vertices by angle equality or orientation).
Choose the appropriate similarity criterion (SSS, SAS, AA) and verify it.
Write proportional relations between corresponding sides and solve for unknown lengths.
If the problem involves area or volume, convert linear scale to area/volume scale (area ∝ k^2, volume ∝ k^3).

Right-triangle similarity and three standard formulas

Context: In a right triangle ABC with right angle at A, let the altitude from A meet the hypotenuse BC at H. The three right triangles △ABC, △ABH, and △ACH are similar. From their similarity we get three useful relations:

AB^2 = BC × BH (the square of leg AB equals the hypotenuse BC times the adjacent hypotenuse segment BH)
AC^2 = BC × CH (the square of leg AC equals the hypotenuse BC times the adjacent hypotenuse segment CH)
AH^2 = BH × CH (the square of the altitude to the hypotenuse equals the product of the two segments BH and CH)

These are standard results derived directly from triangle similarity and are useful to memorize for quick problem solving.

Tip: When given a right triangle with an altitude to the hypotenuse, identify BH and CH (the projections of the legs onto the hypotenuse) and check whether one of the three formulas applies.

Tips and common pitfalls

Always match corresponding vertices in the same order when writing similarity statements.
Use angle equalities to identify correspondence when vertex order is not obvious.
Express the scale factor in simplest terms when required.
In right-triangle problems, look for one of the three altitude relations — many problems reduce to them.

Speaker / source

Lesson presented by a single teacher (self-introduced as Lee Sem; transcription variants possible). Background music and automated subtitles were present; no other speakers featured.