14
Test your knowledge
Textbook: pp. 44–47
GoalConsolidate the first part of the chapter on similarity and test yourself (definition, tests, ratio of areas).
New words
similarity tests · o‘xshashlik alomatlaricoefficient · koeffitsiyentratio of areas · yuzlar nisbaticorresponding elements · mos elementlar
Explanation
A brief recap. Two polygons are similar if corresponding angles are equal and corresponding sides are proportional. Sufficient conditions for triangles are AA, SAS and SSS. Right triangles have three more convenient tests. In similar figures the ratios of perimeters, heights and medians equal k, and the ratio of areas is k². A bisector divides the opposite side proportionally to the adjacent sides. In test questions first decide which test applies, then do only the needed calculation. Typical errors: mixing up k and k², and pairing sides incorrectly.
Worked examples
Test: a triangle similar to one with sides 5, 6, 7 has perimeter 54. k = 54/18 = 3, so the largest side is 21.
Test: for similar triangles with areas 20 and 45, k² = 9/4 and k = 3/2.
Class activity
“Test yourself”: answer 6 test questions in 10 minutes, then swap with a partner, check and write the reasons for mistakes.
Practice
1
Similar triangles have areas 8 and 72. What is the ratio of corresponding sides (smaller to larger)?
1/3
2
ABC ∽ DEF, AB = 12, DE = 8, BC = 15. How long is EF?
10
3
Which test applies to triangles with sides 3, 4, 6 and 9, 12, 18?
SSS: 9/3 = 12/4 = 18/6 = 3.
4
Why does the area grow 9 times rather than 3 times when k = 3?
Area is two-dimensional: both dimensions grow 3 times, 3 · 3 = 9.