01
Create and label the two descendants
Call the original triangle T0. Scaling by two gives T1 with vertices (0, 0), (4, 0) and (0, 2). Translating T1 by (3, 0) gives T2 with vertices (3, 0), (7, 0) and (3, 2). The lineage is T0 to T1 to T2. Each arrow has one named operation, and T2 refers to T1 as its immediate parent. This record prevents the translation from being mistakenly applied to the original coordinates while still labelling the result as a descendant of the scaled triangle. The intermediate coordinates are part of the evidence for that lineage.
02
Compare lengths and areas across generations
T0 has base length two, height one and area one square unit. T1 has base four, height two and area four. The scale factor two doubles corresponding lengths and multiplies area by four. T2 keeps T1’s lengths and area because translation changes position without changing shape or size. These measurements provide checks on the coordinates. If the translated triangle has a different area, the problem is in the recorded points or calculation, not an expected consequence of moving the triangle to the right.
03
Keep a transformation lineage record
Use columns for shape identifier, parent identifier, operation, parameters, resulting vertices and a check. For T1 the check can be the doubled base and height; for T2 it can be the unchanged dimensions and the three-unit shift of every x-coordinate. Preserve the parent record rather than replacing it with the newest shape. A later branch can then start from T0 or T1 explicitly. This is the practical value of the family metaphor: related variants remain traceable even when a collection develops several branches.