In Lesson 25, students discover that the values of the ratios in a right triangle depend solely on the measure of the acute angle by which the adjacent, opposite, and hypotenuse sides are identified. To do this, students first learn how to identify these reference labels. Then, two groups take measurements and make calculations of the values of the and ratios for two sets of triangles, where each triangle in one set is similar to a triangle in the other. This exploration leads to the conclusion regarding the “incredibly useful ratios.”
Incredibly Useful Ratios
Use the right triangle △ 𝐴𝐵𝐶 to answer Exercises 1–3.
For each exercise, label the appropriate sides as adjacent, opposite, and hypotenuse, with respect to the marked acute angle.
Note: Angle measures are approximations.
For each triangle in your set, determine missing angle measurements and side lengths. Side lengths should be measured to one decimal place. Make sure that each of the adj/hyp and opp/hyp ratios are set up and missing values are calculated and rounded appropriately.
With a partner, discuss what you can conclude about each pair of triangles between the two sets.
For each question, round the unknown lengths appropriately. Refer back to your completed chart from the Exploratory Challenge; each indicated acute angle is the same approximated acute angle measure as in the chart. Set up and label the appropriate length ratios, using the terms opp, adj, and hyp in the setup of each ratio.
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