Question

Triangle XYZ with Point X at (-2,4) was dilated with the center at the origin to result in Point x'at(-6,12) Triangle X'Y'Z' was translated 3 units vertically to result in triangle X''Y''Z'' Let p represent the perimeter, in units, of Triangle XYZ. Which expression represents the perimeter, in units, of Triangle X''Y''Z'' ?

Answer 1

To find the perimeter of Triangle X''Y''Z'', multiply the perimeter of the original Triangle XYZ (represented by 'p') by the scalar dilation factor, which is 3. Thus, the perimeter of Triangle X''Y''Z'' is expressed as 3p.

Step-by-step explanation:

The student's question asks how to find the expression representing the perimeter of Triangle X''Y''Z'', given the transformations that have occurred starting with Triangle XYZ. Since Triangle XYZ was dilated with the origin as the center to form Triangle X'Y'Z' and then translated to form Triangle X''Y''Z'', the size of the triangle has not changed due to the translation. The dilation increased the size of the triangle by the same scale factor applied to Point X to obtain Point X', which is a factor of 3, as seen from the coordinates moving from (-2,4) to (-6,12). Therefore, the perimeter of Triangle X''Y''Z'' is 3 times the perimeter of the original Triangle XYZ.

To represent this mathematically, if 'p' is the perimeter of Triangle XYZ, then the perimeter of Triangle X''Y''Z'' would be 3p.