Question

question 2 points 3
which of the following is true for two non - congruent triangles δxyz
and δpqr?

a translation can make δxyz and δpqr congruent.

a rotation can make δxyz and δpqr congruent.

dilation and compression can make δxyz and δpqr congruent.

a glide reflection can make δxyz and δpqr congruent.

Explanation:

Step1: Recall congruence and transformations

Congruent triangles have same shape and size. Translations, rotations, glide reflections are rigid transformations (preserve size and shape), so they can only map a triangle to a congruent one. Non - congruent triangles have different sizes (or shapes, but here since they are triangles, different sizes as shape is triangle). Dilation/compression changes size (scales the figure). If two triangles are non - congruent (say different side lengths), dilation (scaling up/down) and compression (scaling in the other way) can adjust the size to make them congruent (same size and shape, since they are triangles, same angles maybe? Wait, no, for triangles, if we can scale one to match the other in size, and since they are triangles, if angles are same, but actually, the key is: rigid transformations preserve size, so they can't make non - congruent triangles congruent. Dilation/compression can change size. So if two triangles are similar (same shape, different size), dilation/compression can make them congruent. Since the triangles are non - congruent, they could be similar (different size, same shape). So dilation and compression (which are similarity transformations that can adjust size) can make them congruent. The other options: translation, rotation, glide reflection are rigid (preserve size), so they can't make non - congruent (different size) triangles congruent.

Step2: Analyze each option

• Option 1: Translation is rigid, preserves size. If triangles are non - congruent (different size), translation can't make them congruent. Eliminate.
• Option 2: Rotation is rigid, preserves size. Can't make non - congruent triangles congruent. Eliminate.
• Option 3: Dilation (scaling) and compression (scaling in reverse) can change the size of a triangle. If two triangles are similar (same shape, different size), dilation/compression can make their sizes equal, hence congruent. This is correct.
• Option 4: Glide reflection is rigid, preserves size. Can't make non - congruent triangles congruent. Eliminate.

Answer:

Dilation and compression can make $\triangle XYZ$ and $\triangle PQR$ congruent. (The option with "Dilation and compression can make $\triangle XYZ$ and $\triangle PQR$ congruent.")