To determine if two triangles are necessarily congruent when points are moved, we need to consider the definition of congruence. Two triangles are congruent if they have the same shape and size, meaning their corresponding sides and angles are equal.
When you have a triangle formed by points A, B, and C, and then a point F is moved around, changing the shape or dimensions of the triangle can affect the congruence depending on how points A, B, C, and F are manipulated.
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Moving Point F: If point F is moved such that it forms a new triangle with points A, B, and C, the newly formed triangle can only be congruent to the original triangle if the conditions of congruence are satisfied (SSS, SAS, ASA, AAS, or HL depending on the configuration of the triangles).
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Moving Points A, B, or C: When A, B, or C are moved to resize the first triangle, this can also lead to changes in the angles and sides of the triangle. If these points are moved while maintaining the relative lengths of the sides and angles, the original triangle can be re-scaled but will not necessarily maintain congruence with the triangle formed with point F unless precise measures are taken.
In conclusion, the triangles are not necessarily congruent unless specific conditions are met concerning the lengths of the sides and the measures of the angles. Simply moving the points around does not guarantee that the resulting triangles will be congruent, as congruence requires equal corresponding sides and angles.