Assess two figures using the definitions of transformations to determine whether the figures are similar. Is ΔFGH∼ΔIJK, and for what reason?

Question

No. If point J was at (6,-6), the triangles would be similar
No, if point J was at (-6,6), the triangles would be similar
Yes. All sides are proportional
Yes. All angles are congruent

Answer 1

To determine whether the triangles ΔFGH and ΔIJK are similar, we need to assess the criteria for triangle similarity. The main conditions for two triangles to be similar are:

1. Angle-Angle (AA) Criterion: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. Side-Side-Side (SSS) Criterion: If the corresponding sides of two triangles are proportional, then the triangles are similar.
3. Side-Angle-Side (SAS) Criterion: If one angle of one triangle is congruent to one angle of another triangle, and the sides including those angles are in proportion, then the triangles are similar.

From the options you provided, let's analyze the reasoning:

1. "No. If point J was at (6,-6), the triangles would be similar." - This statement is a conditional suggestion and does not address the current similarity without the hypothetical change.

2. "No, if point J was at (-6,6), the triangles would be similar." - Similar to the first option, this is a conditional statement that does not substantiate the current figures' similarity.

3. "Yes. All sides are proportional." - If it can be established that all corresponding sides are proportional, then this statement confirms that the triangles are similar based on the SSS criterion.

4. "Yes. All angles are congruent." - If it can be shown that all corresponding angles are congruent, this confirms similarity based on the AA criterion.

To accurately determine the similarity of the triangles ΔFGH and ΔIJK, we would need to examine the actual lengths of the sides and the measures of the angles. Without that specific information, we cannot definitively conclude whether the triangles are similar.

However, if you know either all angles are congruent or all sides are proportional, you can use those facts to determine the similarity. Based on the typical understanding of triangle similarity, the clearer affirmative answers are options 3 and 4, depending on the given conditions:

• If all sides are proportional, then "Yes. All sides are proportional" is the reason for similarity.
• If all angles are congruent, then "Yes. All angles are congruent" is the reason for similarity.

Ultimately, the correct answer will depend on the specific comparative information about the sides or angles of the triangles in question.