Question

triangle xyz was translated up 2 and right 3 units, creating triangle cab. which relationship between the parts of the two triangles must be true?
○ ( angle x cong angle a )
○ ( overline{zx} cong overline{ca} )
○ ( overline{yz} cong overline{ab} )
○ ( angle z cong angle c )

Explanation:

Step1: Recall Translation Properties

Translation is a rigid transformation, so corresponding sides and angles of the pre - image (triangle XYZ) and image (triangle CAB) are congruent. We need to identify the corresponding parts.
Looking at the triangles, the correspondence should be: X corresponds to C, Y corresponds to A, Z corresponds to B (since triangle XYZ is translated to triangle CAB). Wait, no, let's check the labels again. The pre - image is XYZ and the image is CAB. So vertex X corresponds to vertex C, Y corresponds to A, Z corresponds to B? Wait, no, maybe the order is XYZ to CAB, so X→C, Y→A, Z→B? Wait, no, let's think about the translation. When we translate a figure, the corresponding vertices are related by the translation vector. But also, the sides: in triangle XYZ, side ZX is between Z and X, and in triangle CAB, side CA is between C and A. Wait, if Y corresponds to A, Z corresponds to B, X corresponds to C. Then side ZX (from Z to X) and side CA (from C to A)? Wait, no, let's list the corresponding parts properly.

Wait, the pre - image is XYZ, image is CAB. So vertex X corresponds to vertex C, Y corresponds to A, Z corresponds to B. So angle at X (∠X) corresponds to angle at C (∠C), angle at Y (∠Y) corresponds to angle at A (∠A), angle at Z (∠Z) corresponds to angle at B (∠B). Side XY corresponds to side CA, side YZ corresponds to side AB, side ZX corresponds to side BC? Wait, no, maybe I got the correspondence wrong. Wait, let's look at the labels of the triangles. The first triangle (pre - image) has vertices Y, Z, X. The second (image) has A, B, C. When we translate up 2 and right 3, the correspondence should be Y→A, Z→B, X→C. So:

- ∠Y corresponds to ∠A
- ∠Z corresponds to ∠B
- ∠X corresponds to ∠C
- Side YZ corresponds to side AB
- Side ZX corresponds to side BC
- Side XY corresponds to side AC

Wait, now let's check the options:

Option 1: ∠X ≅ ∠A. But ∠X corresponds to ∠C, ∠A corresponds to ∠Y. So this is false.

Option 2: \(\overline{ZX}\cong\overline{CA}\). \(\overline{ZX}\) is from Z to X, \(\overline{CA}\) is from C to A. If Z→B, X→C, A→Y, then \(\overline{ZX}\) and \(\overline{CA}\) are not corresponding sides. Wait, maybe my correspondence is wrong. Let's try another way. Maybe the triangle XYZ is translated to CAB, so X→C, Y→A, Z→B. So:

- Vertex X → C
- Vertex Y → A
- Vertex Z → B

Then side ZX: Z to X, which is B to C. Side CA: C to A, which is X to Y. No, that's not right. Wait, maybe the labels of the image triangle are C, A, B, so the triangle is CAB, so the vertices are C, A, B in order. The pre - image is X, Y, Z in order (XYZ). So the translation maps X to C, Y to A, Z to B. So:

- ∠X (at X) corresponds to ∠C (at C)
- ∠Y (at Y) corresponds to ∠A (at A)
- ∠Z (at Z) corresponds to ∠B (at B)
- Side XY (between X and Y) corresponds to side CA (between C and A)
- Side YZ (between Y and Z) corresponds to side AB (between A and B)
- Side ZX (between Z and X) corresponds to side BC (between B and C)

Now let's check the options:

Option 1: ∠X ≅ ∠A. ∠X corresponds to ∠C, ∠A corresponds to ∠Y. So false.

Option 2: \(\overline{ZX}\cong\overline{CA}\). \(\overline{ZX}\) corresponds to \(\overline{BC}\), \(\overline{CA}\) corresponds to \(\overline{XY}\). So false.

Option 3: \(\overline{YZ}\cong\overline{AB}\). \(\overline{YZ}\) is between Y and Z, \(\overline{AB}\) is between A and B. Since Y→A and Z→B, \(\overline{YZ}\) and \(\overline{AB}\) are corresponding sides. So by the property of translation (rigid transformation, corresponding sides are congruent), \(\overline{YZ}\cong\overline{AB}\) is true.

Option 4: ∠Z ≅ ∠C. ∠Z corresponds to ∠B, ∠C corresponds to ∠X. So false.

Answer:

\(\overline{YZ}\cong\overline{AB}\) (the option with \(\overline{YZ}\cong\overline{AB}\))