Triangle

Question

is isosceles with base angles
and
.

The figure presents isosceles triangle A B C. Base A C is horizontal, and vertex B is above base A C. Side A B is labeled 3 and side B C is labeled 3. Point E is indicated on base A C. Line segment B E is drawn such that it extends above B and extends below E and is labeled m. A point in the middle of segment B E is labeled D.
In triangle
, the length of

is not equal to
, point
is the midpoint of

, and line
bisects

at point
. Which of the following transformations can be used to prove that
and
are congruent?

Responses

Rotating triangle

clockwise about point

Answer A: Rotating triangle A B C 60 degrees clockwise about point D
A

Reflecting triangle
over line

Answer B: Reflecting triangle A B C over line m
B

Translating triangle
along line segment

Answer C: Translating triangle A B C along line segment A C
C

Reflecting triangle
over line segment

Answer 1

To determine which transformation can be used to prove that the two triangles in question are congruent, we need to analyze the given options based on the properties of isosceles triangles and the characteristics of congruence.

1. Rotating triangle A B C 60 degrees clockwise about point D: This transformation implies turning the triangle around point D, which is the midpoint of segment BE. However, this rotation does not guarantee that it will match the position of another triangle unless specifically stated. So, this option is not a conclusive transformation for proving congruence.

2. Reflecting triangle A B C over line m: This option implies that there is a line of reflection (line m) that can create a mirror image of triangle A B C resulting in a triangle that is congruent to it. Given that reflection preserves distances and angles, this could indeed lead to congruence if line m is appropriately positioned.

3. Translating triangle A B C along line segment A C: This would mean sliding the triangle along the base AC. This transformation does not change the orientation or flip the triangle, and therefore may not necessarily produce a triangle that is congruent in a different position unless the triangles are already aligned.

4. Reflecting triangle over line segment: The answer seems to be cut off, but if this option suggests reflecting over a line that bisects one of the angles or segments involving triangle A B C, then it could imply congruence, but without the full context, it’s hard to interpret.

Given these analyses, the strongest option likely to prove the congruence of the two triangles in question, especially considering the characteristics of isosceles triangles (where angles and sides are equal), would be:

Answer B: Reflecting triangle A B C over line m.

This transformation would ensure that triangle A B C has a congruent counterpart after the reflection, as it would maintain all lengths and angles, consistent with the properties of triangle congruence.