Lesson 4
Ratios in Right Triangles
 Let’s investigate ratios in the side lengths of right triangles.
4.1: Ratio Rivalry
Consider \(\frac{a}{c} \text{ and } \frac{b}{d}\). Which is greater, or are they equal? Explain how you know.
4.2: Tons of Triangles
Your teacher will give you some angles.
 Use the applet to build 4 different right triangles for each of your angles.
 Record the side lengths of each of the triangles.
 Compute these 3 quotients for the acute angles in each triangle:
 The length of the leg adjacent to your angle divided by the length of the hypotenuse
 The length of the leg opposite from your angle divided by the length of the hypotenuse
 The length of the leg opposite from your angle divided by the length of the leg adjacent to your angle
 Find the mean of each type of quotient.
 What do you notice? What do you wonder?
4.3: Tons of Ratios
 Compare the row for 20 degrees and the row for 70 degrees in the right triangle table. What is the same? What is different?
 The row for 55 degrees is given here. Complete the row for 35 degrees.
angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg \(35^\circ\) \(55^\circ\) 0.574 0.819 1.428  What do you know about a triangle with an adjacent leg to hypotenuse ratio value of 0.839?
 What is the range for the possible ratios of each of the following ratios?
 adjacent leg \(\div\) hypotenuse
 opposite leg \(\div\) hypotenuse
 opposite leg \(\div\) adjacent leg
 What would the triangle look like if the adjacent leg \(\div\) hypotenuse ratio was 1? Greater than 1?
Summary
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.
These triangles are all similar by the AngleAngle Triangle Similarity Theorem. Focusing on the 25 degree angles, we see that all 3 triangles have adjacent leg to hypotenuse ratios of approximately 0.91.
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 right triangle table comes from measuring and finding ratios in several right triangles with different angle measures.
angle  adjacent leg \(\div\) hypotenuse  opposite leg \(\div\) hypotenuse  opposite leg \(\div\) adjacent leg 

\(25^\circ\)  0.906  0.423  0.466 
\(35^\circ\)  0.819  0.574  0.700 
\(45^\circ\)  0.707  0.707  1.000 
\(55^\circ\)  0.574  0.819  1.428 
\(65^\circ\)  0.423  0.906  2.145 
Some ratios in this table are repeated. Notice that the rows for 25 degrees and 65 degrees have 2 of the same ratios. What is special about 25 and 65? They are complementary angles, that is, the 2 angles sum to 90 degrees. This seems to be true for other complementary angles. Notice that \(35+55=90\) and those rows both have 0.819 as a ratio.
Glossary Entries

complementary
Two angles are complementary to each other if their measures add up to \(90^\circ\). The two acute angles in a right triangle are complementary to each other.