Unit 4, Lesson 4
Ratios in Right Triangles
Geometry
4.1
Warm-up
Consider . Determine which ratio is greater or whether they are equal. Explain how you know.
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Consider . Determine which ratio is greater or whether they are equal. Explain how you know.
All right triangles that contain the same acute angles are similar to each other. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles.
Three right triangles, On left triangle ABC, in middle triangle DEF, on right triangle LMN. AB=9 point 1 units, BC= 10 units, CA=4 point 2 units. DE =6 point 5 units, EF= 7 point 1 units, FD= 3 units. LM=5 point 4 units, MN= 5 point 9 units,NL =2 point 5 units. Angles A,D,L all = 90 degrees. Angles B,E,M all = 25 degrees.
Because all right triangles with the same acute angle measures have the same ratios, we can look for patterns that will help us solve problems. The values in this Right Triangle Table come from measuring triangles very precisely and then finding ratios of the sides. The results in the table are written out to the thousandths place, which is more accurate than any ratio we could calculate when measuring by hand.
| angle | adjacent leg hypotenuse | opposite leg hypotenuse | opposite leg adjacent leg |
|---|---|---|---|
| 0.906 | 0.423 | 0.466 | |
| 0.819 | 0.574 | 0.700 | |
| 0.707 | 0.707 | 1.000 | |
| 0.574 | 0.819 | 1.428 | |
| 0.423 | 0.906 | 2.145 |
Some ratios in this table are repeated. Notice that the rows for 25 degrees and 65 degrees have two of the same ratios. What is special about 25 and 65? They are complementary angles, that is, two angles sum to 90 degrees. This seems to be true for other complementary angles. Notice that and those rows both have 0.819 as a ratio.