Lecture Notes: Finding Ratios in Similar Right Triangles

Understanding Similar Right Triangles

Definition: Similar triangles are triangles that have the same shape but may differ in size.
Characteristics:
- Corresponding angles are equal.
- Corresponding sides are proportional.

Proportional Sides: The sides of similar triangles are proportional. This means if triangle ABC is similar to triangle DEF, then:
- (\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF})
Finding Ratios:
- To find the ratio of sides, compare corresponding sides.
- Example: If triangle ABC is similar to triangle DEF, and (AB = 3, DE = 6), then the ratio (\frac{AB}{DE} = \frac{3}{6} = \frac{1}{2}).

Steps:
1. Identify the similar triangles.
2. Identify corresponding sides and angles.
3. Set up the proportion based on corresponding sides.
4. Solve for the unknown side or ratio.

Problem 1:
- Given two similar triangles where one triangle has sides 3, 4, 5 and the other has a hypotenuse of 10, find the lengths of the other sides of the second triangle.
- Solution:
  1. Use the ratio (\frac{5}{10} = \frac{1}{2}).
  2. Apply it to other sides:
    - Smaller side 3 corresponds to (3 * 2 = 6).
    - Other side 4 corresponds to (4 * 2 = 8).
Problem 2:
- Determine the ratio of sides for two similar triangles with sides labeled as 7, 24, 25 and 14, x, 50 respectively.
- Solution:
  1. Set up ratios: (\frac{7}{14} = \frac{1}{2}) for the first pair of sides.
  2. Apply ratio to find x: (\frac{24}{x} = \frac{1}{2}) gives (x = 48).

Similar triangles have proportional corresponding sides.
Always ensure angles are corresponding and equal to confirm similarity.
Use proportions to find unknown sides or angles in problems involving similar triangles.