Triangle ABC is similar to triangle A'B'C'. Select all statements that must be true.
Understand the Problem
The question involves determining which statements are true based on the similarity of triangles ABC and A'B'C'. This can be approached by using the properties of similar triangles, particularly the corresponding sides and angles.
Answer
A, B
Answer for screen readers
The true statements are A and B.
Steps to Solve
- Understanding Triangle Similarity
Triangles ABC and A'B'C' are similar, which means all corresponding angles are equal and the ratios of the lengths of corresponding sides are proportional.
- Identifying Corresponding Sides for Ratios
For similar triangles, we can establish the following relationships for corresponding sides:
- $ \frac{AB}{A'B'} = \frac{BC}{B'C'} = \frac{CA}{C'A'} $
- Analyzing Each Statement
We will analyze each statement based on the properties of similar triangles:
- Statement A: $ \frac{BC}{B'C'} = \frac{AC}{A'C'} $
This statement is true as it directly represents one of the characteristics of similar triangles.
- Statement B: $ \frac{AB}{A'B'} = \frac{CA}{C'A'} $
This statement is also true by the similarity property.
- Statement C: $ \frac{B'C'}{BA} = \frac{B'A'}{BC} $
This statement rearranges the ratios and is not generally valid; it's not necessarily a true statement.
- Statement D: $ \frac{AB}{B'C'} = \frac{BC}{A'B'} $
This statement compares sides in a way that doesn't conform to the similarity rules; hence, it's false.
- Statement E: $ \frac{AC}{BC} = \frac{A'C'}{B'C'} $
This statement rearranges ratios that do not align with the property of similar triangles; thus, it's false.
- Conclusion of Statements
Based on the analysis:
- A and B are true.
- C, D, and E are false.
The true statements are A and B.
More Information
In geometry, similar triangles maintain a consistent ratio among their corresponding sides. The properties of similarity are foundational in understanding how shapes relate to one another. This principle also aids in solving various geometric problems, such as finding unknown side lengths or angles based on given measurements.
Tips
- Confusing the order of the vertices when applying the similarity ratio.
- Forgetting that all corresponding angles must also be equal for triangles to be similar.
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