Question

is \\( \triangle a b c \\) congruent to \\( \triangle x y z \\)? justify your answer using rigid motions.
yes. a translation and a reflection can map \\( \triangle a b c \\) onto \\( \triangle x y z \\).
yes. a reflection and a dilation can map \\( \triangle a b c \\) onto \\( \triangle x y z \\).
no. it is not possible to map \\( \triangle a b c \\) onto \\( \triangle x y z \\) using only rigid motions.

Explanation:

Step1: Recall the definition of rigid motions

Rigid motions (translations, rotations, reflections) preserve the size and shape of a figure.

Step2: Check the side - length ratios

For \(\triangle ABC\) with sides \(a = 6\), \(b = 8\), \(c = 10\) and \(\triangle XYZ\) with sides \(a'=9\), \(b' = 12\), \(c'=15\).
The ratio of corresponding sides: \(\frac{6}{9}=\frac{2}{3}\), \(\frac{8}{12}=\frac{2}{3}\), \(\frac{10}{15}=\frac{2}{3}\). But dilation (a non - rigid motion as it changes the size) is needed to get from \(\triangle ABC\) to \(\triangle XYZ\) since the side lengths are in proportion but not equal. Rigid motions do not change the size of a figure.

Answer:

No. It is not possible to map \(\triangle ABC\) onto \(\triangle XYZ\) using only rigid motions.