19
Properties of similar polygons
Textbook: pp. 60–61
GoalKnow and apply the properties of similar polygons (perimeter ratio k, area ratio k², division into the same number of similar triangles).
New words
ratio of perimeters · perimetrlar nisbatiratio of areas · yuzlar nisbatidiagonal · diagonaldividing into similar triangles · o‘xshash uchburchaklarga ajratish
Explanation
The ratio of perimeters of similar polygons equals the similarity coefficient k, since every side is multiplied by k: P₁ = k·P. Diagonals from one vertex divide similar polygons into the same number of pairwise similar triangles, the key helper fact in the proof. The ratio of the areas of each pair is k², so the ratio of the areas of the whole polygons is also k²: S₁ = k²·S. In a problem first find k from perimeters or corresponding sides, then use k² for areas. A triangle and a quadrilateral can never be similar because they have different numbers of angles.
Worked examples
Similar polygons with perimeters 20 and 35: k = 35/20 = 7/4, and the ratio of areas is 49/16.
Similar polygons have areas 12 and 27: k² = 27/12 = 9/4, so k = 3/2. The side matching an 8 cm side of the smaller is 8 · 3/2 = 12 cm.
Class activity
“Five corners”: divide a pentagon into 3 triangles with diagonals from one vertex; do the same in a similar pentagon and compare the pairs of triangles.
Practice
1
Similar polygons have perimeters 30 and 45. If the smaller has area 40, what is the larger's area?
90
2
Similar polygons have areas in the ratio 4 : 9, and the larger area exceeds the smaller by 35. Find the areas.
28 and 63
3
If k = 1/3, what is the ratio of the smaller area to the larger?
1/9
4
Why can a triangle and a quadrilateral never be similar?
Similar figures need equal corresponding angles, which requires the same number of angles.