Lesson 2 — Similar Figures
Big idea: Similar figures have the same shape: corresponding angles are equal and corresponding sides are proportional; a dilation produces a similar copy from a center point.
Words to know
| Word | Meaning |
|---|---|
| Similar figures | Figures with the same shape, so corresponding angles are equal and corresponding sides are proportional. |
| Similar triangles | Triangles for which a vertex correspondence makes corresponding sides proportional and corresponding angles equal. |
| Similarity correspondence | The matching of vertices, sides, and angles; the order in △ABC ~ △DEF tells you which parts match. |
| Similarity | A correspondence between points such that corresponding segments are proportional. |
| Dilation | A transformation that enlarges or reduces a figure from a center point by a fixed ratio. |
| Center of a dilation | The fixed point O from which all image points are measured. |
| Magnitude | The positive number r giving the ratio of image length to original length. |
Concepts to know
- Definition: Two triangles are similar iff there is a correspondence between their vertices such that corresponding sides are proportional and corresponding angles are equal.
- Write △ABC ~ △DEF only with vertices in matching order. From that statement you can read off equal angles and proportional sides.
- A dilation with center O and magnitude r sends each point P to P′ with OP′ = r·OP.
- If r > 1, the image is larger; if 0 < r < 1, it is smaller; if r = 1, it is the same size.
- Similarity preserves shape and angles, but not necessarily size.
- The definition extends beyond triangles to other figures, such as rectangles and polygons.
Formulas / rules
- Corresponding sides: AB/DE = BC/EF = AC/DF
- Corresponding angles: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
- Dilation: OP′ = r · OP
- Magnitude: r = (image length)/(original length)
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Name corresponding parts | Match the vertex order in the similarity statement. |
| Find a missing side | Write a proportion using corresponding sides and solve. |
| Find a missing angle | Use the equal corresponding angles; angles do not change under similarity. |
| Find the magnitude of a dilation | Divide an image length by the matching original length. |
| Draw a dilation image | Draw rays from the center through each vertex, then mark P′ so OP′ = r·OP. |
| Check whether two figures are similar | Check equal angles and proportional sides. |
Watch out
- The order of vertices in a similarity statement is not optional.
- Proportional sides are not necessarily equal; equal sides would make the figures congruent.
- A dilation changes lengths but not angle measures.
- The magnitude r must be positive; check whether the image is larger or smaller than 1.
- Similar is not the same as congruent unless r = 1.