Question

the translation \\((x - 10, y + 17)\\) is applied to a triangle. maryanne makes a conjecture about the perimeter of the image of the triangle, tests the conjecture, and finds that it is true. what could have been her conjecture? \\(\bigcirc\\) the perimeter of the image is 17 units longer than the perimeter of the original triangle. \\(\bigcirc\\) the perimeter of the image is 10 units shorter than the perimeter of the original triangle. \\(\bigcirc\\) the perimeter of the image is the same as the perimeter of the original triangle. \\(\bigcirc\\) the perimeter of the image is 10 times the perimeter of the original triangle.

Explanation:

A translation in geometry is a rigid transformation, which means it only moves the figure (in this case, the triangle) without changing its shape, size, or the lengths of its sides. Since the perimeter of a triangle is the sum of the lengths of its three sides, and a translation doesn't alter the side lengths, the perimeter of the image (the translated triangle) should be the same as the perimeter of the original triangle. The other options suggest changes in perimeter (either by a fixed length or a multiple), which are not characteristics of a translation.

Answer:

The perimeter of the image is the same as the perimeter of the original triangle.