10
The third test for similar triangles
Textbook: pp. 36–37
GoalKnow the third similarity test (SSS) and decide similarity from three proportional sides.
New words
SSS test (side-side-side) · TTT alomatithree sides · uchta tomonratio of sides · tomonlar nisbatismallest side · kichik tomon
Explanation
Third test (SSS): if three sides of one triangle are proportional to three sides of another, the triangles are similar. To check, list the sides in increasing order and divide smallest by smallest, middle by middle, largest by largest; if all three ratios are equal, the triangles are similar. This test is used when the angles are unknown. Once similarity is known, corresponding angles are equal, which is useful in problems. If one of the ratios differs from the others, the triangles are not similar.
Worked examples
3, 4, 5 and 9, 12, 15: 9/3 = 12/4 = 15/5 = 3, so they are similar with k = 3.
5, 7, 8 and 10, 14, 17: 10/5 = 2 and 14/7 = 2 but 17/8 ≠ 2, so the triangles are not similar.
Class activity
“Triple cards”: each student gets a card with three numbers; the class finds “families” of similar triangles and states each family's coefficient.
Practice
1
Are triangles with sides 8, 10, 12 and 12, 15, 18 similar? What is k?
Yes; 12/8 = 15/10 = 18/12 = 3/2.
2
A triangle similar to one with sides 6, 9, 12 has smallest side 8. How long is its largest side?
16
3
Are triangles with sides 7, 9, 11 and 14, 18, 24 similar?
No: 14/7 = 18/9 = 2 but 24/11 ≠ 2.
4
Why must the sides be compared in increasing order in the SSS test?
To pair corresponding sides correctly: the smallest matches the smallest and the largest matches the largest.