Question 15 of 50advanced🔍 AnalyzeShort Answer2 marks

The circumcircle of a given ΔABC\Delta ABC is drawn. Can there be other triangles congruent to ΔABC\Delta ABC that share the same circumcircle?

Correct Answer

Yes, there can be other triangles congruent to ΔABC\Delta ABC that share the same circumcircle. The sides of ΔABC\Delta ABC are chords of the circumcircle. As shown in the context, a circle can have multiple equal chords (e.g., AB = DE).

By drawing a set of chords on the same circumcircle that are equal in length to the corresponding sides of ΔABC\Delta ABC, we can form a new inscribed triangle. Since all corresponding sides of this new triangle are equal to those of ΔABC\Delta ABC, the two triangles will be congruent by the SSS congruence rule. Thus, other congruent triangles can share the same circumcircle.

Exercise: Think, Draw and Infer | Q: 2 | (Page 7)
For More Understanding

Explanation

The textbook context establishes that a circle can have equal chords (like AB = DE) and applies the SSS congruence rule to triangles formed by the center and these chords. Extending this logic to inscribed triangles, if we construct a new triangle on the same circumcircle using chords equal in length to the sides of the original triangle, the corresponding sides of the two inscribed triangles will be equal. Therefore, by the SSS congruence criterion, the new inscribed triangle will be congruent to the original triangle, proving that multiple congruent triangles can share the same circumcircle.

Solution Steps

  1. Step 1: Recognize that the sides of an inscribed ΔABC\Delta ABC are chords of the circumcircle, and a circle can have other chords equal in length to these sides (e.g., AB = DE).

  2. Step 2: Conclude that drawing a new set of chords equal to the sides of ΔABC\Delta ABC forms a new inscribed triangle with equal corresponding sides, making it congruent by the SSS congruence rule.

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