Lesson 5

Working with Ratios in Right Triangles

  • Let’s solve problems about right triangles.

5.1: Launch Pad

When a rocket is launched, it climbs 50 feet for every 13 feet it travels horizontally. Draw a diagram to represent the situation. Then estimate the rocket’s launch angle.

5.2: Pythagorean Triples

  1. Sketch the triangle with side lengths 7, 24, and 25 units. Label the smallest angle \(A\).
  2. Find the 3 ratios of side lengths for angle \(A\).
  3. Estimate the acute angles in this triangle.


  1. Find another Pythagorean triple.
  2. Estimate the acute angles in that triangle. Explain how you know.

5.3: Solve All the Triangles

  1. What is the length of segment \(AB\)?

    A right triangle.
  2. In a right triangle with one angle measuring 40 degrees, the leg opposite the 40 degree angle is 5 cm. What is the length of the hypotenuse?

  3. What is the length of segment \(DE\)?

    Right triangle DEF. side EF = 6. Angle F = 50 degrees. 
  4. In a right triangle with one angle measuring 70 degrees, the leg opposite the 70 degree angle is 12 cm. What is the length of the leg adjacent to the 70 degree angle?

Summary

All right triangles that contain the same acute angle are similar. This means that the ratios of corresponding side lengths are equal for all right triangles with the same acute angles. Using the ratios calculated in the previous lesson and properties of similar triangles, we can calculate and estimate unknown side lengths and angles in right triangles.

If we measure the legs of any right triangle with an angle of 25 degrees, the ratio of the leg opposite the 25 degree angle to the leg adjacent to the 25 degree angle will always be 0.466. Therefore, we can find length \(b\). Since \(\frac{5}{b} = 0.466\), we know \(b\) is 10.7 units.

Right triangle ABC. Side BC = 5 units. Angle A = 25 degrees. 

Similarly, we can estimate the measure of the missing angles in triangle \(DEF\).

angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
\(50^\circ\) 0.643 0.766 1.192
\(60^\circ\) 0.500 0.866 1.732

The ratio of the leg opposite angle \(D\) to the hypotenuse is 0.794. This value is between the value of opposite leg divided by hypotenuse for 50 degrees (0.766) and 60 degrees (0.866). So the measure of angle \(D\) must be between 50 and 60 degrees. Similarly, the leg opposite angle \(D\) divided by the leg adjacent to angle \(D\) gives a ratio of 1.283, which is between the same ratio for 50 degrees (1.192) and 60 degrees (1.732). The exact value turns out to be 52 degrees.

Right triangle DEF. DE= 6 units, EF = 7 point 7 units, FD =  9 point 7 units. Right angle E.

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.