11
Tests for similar right triangles
Textbook: pp. 38–39
GoalKnow and apply the three similarity tests for right triangles.
New words
leg · katethypotenuse · gipotenuzaacute angle · o‘tkir burchakright triangle · to‘g‘ri burchakli uchburchak
Explanation
Because each right triangle has a right angle, the similarity tests simplify. 1) If one acute angle of one equals one acute angle of the other, the triangles are similar. 2) If the legs are proportional, they are similar. 3) If the hypotenuse and a leg of one are proportional to the hypotenuse and a leg of the other, they are similar. The first two are special cases of AA and SAS; the third is proved separately using the hypotenuse-leg congruence test. These tests are widely used to compare triangles formed by a height.
Worked examples
Right triangles with legs 3, 4 and 6, 8 have 6/3 = 8/4 = 2, so they are similar; their hypotenuses are 5 and 10.
A triangle with hypotenuse 13 and leg 5, and one with hypotenuse 26 and leg 10: 26/13 = 10/5 = 2, so they are similar; the other legs are 12 and 24.
Class activity
“Paper triangles”: with an adult's help with scissors, cut right triangles of different sizes from squared paper, pair up the similar ones and say which test applies.
Practice
1
A right triangle similar to one with legs 5 and 12 has shorter leg 15. How long is the longer leg?
36
2
In one triangle the hypotenuse is 10 and a leg is 6; in another the hypotenuse is 15 and a leg is 9. Are they similar?
Yes: 15/10 = 9/6 = 3/2 (test 3).
3
Two right triangles each have a 35° angle. Are they similar?
Yes, by test 1: the right angles and the 35° angles are equal.
4
Why is checking one more angle enough for right triangles?
The right angles are already equal, so one more equal angle gives the AA test.