Question
Which of these triangle pairs can be mapped to each other using a reflection and a translation?
Triangles L R K and A R Q are connected at point R. Triangle L R K is reflected across point R to form triangle A R Q.
Triangles L P K and Q R A are shown. Triangle L P K is reflected across a line to form triangle Q R A. Triangle Q R A is shifted to the right of triangle L P K.
Triangles L P K and Q R A are shown. Triangle L P K is shifted down and to the left to form triangle Q R A.
Triangles L P K and Q R A are shown. Triangle L P K is rotated to form triangle Q R A. Triangle Q R A is also shifted to the right.
Triangles L R K and A R Q are connected at point R. Triangle L R K is reflected across point R to form triangle A R Q.
Triangles L P K and Q R A are shown. Triangle L P K is reflected across a line to form triangle Q R A. Triangle Q R A is shifted to the right of triangle L P K.
Triangles L P K and Q R A are shown. Triangle L P K is shifted down and to the left to form triangle Q R A.
Triangles L P K and Q R A are shown. Triangle L P K is rotated to form triangle Q R A. Triangle Q R A is also shifted to the right.
Answers
GPT-4o mini
To determine which triangle pairs can be mapped to each other using a reflection and a translation, we need to analyze the given scenarios:
1. **Triangles L R K and A R Q**: It is stated that triangle L R K is reflected across point R to form triangle A R Q. This process involves a reflection and suggests that the two triangles are congruent. Therefore, this pair can be mapped to each other using reflection and translation.
2. **Triangles L P K and Q R A**: Here, it states that triangle L P K is reflected across a line to form triangle Q R A. This is also a reflection. If triangle Q R A is then simply shifted to the right of triangle L P K, this combination of reflection and translation makes it possible for triangle L P K to be mapped to triangle Q R A.
3. **Triangles L P K and Q R A (shifted down and to the left)**: In this case, triangle L P K is said to be shifted down and to the left to form triangle Q R A. This is a translation only, without a reflection involved. Therefore, they are not related in the manner described (reflection followed by translation).
4. **Triangles L P K and Q R A (rotation and translation)**: Here, triangle L P K is rotated to form triangle Q R A and then shifted to the right. Since rotation is not a reflection, this scenario does not fit the criteria of being mapped by reflection and then translation.
Given the analysis, the triangle pairs that can be mapped to each other using a reflection and a translation are:
- Triangles L R K and A R Q (reflected across point R).
- Triangles L P K and Q R A (reflected across a line and then translated).
The third and fourth scenarios do not meet the requirements of a reflection followed by a translation.
1. **Triangles L R K and A R Q**: It is stated that triangle L R K is reflected across point R to form triangle A R Q. This process involves a reflection and suggests that the two triangles are congruent. Therefore, this pair can be mapped to each other using reflection and translation.
2. **Triangles L P K and Q R A**: Here, it states that triangle L P K is reflected across a line to form triangle Q R A. This is also a reflection. If triangle Q R A is then simply shifted to the right of triangle L P K, this combination of reflection and translation makes it possible for triangle L P K to be mapped to triangle Q R A.
3. **Triangles L P K and Q R A (shifted down and to the left)**: In this case, triangle L P K is said to be shifted down and to the left to form triangle Q R A. This is a translation only, without a reflection involved. Therefore, they are not related in the manner described (reflection followed by translation).
4. **Triangles L P K and Q R A (rotation and translation)**: Here, triangle L P K is rotated to form triangle Q R A and then shifted to the right. Since rotation is not a reflection, this scenario does not fit the criteria of being mapped by reflection and then translation.
Given the analysis, the triangle pairs that can be mapped to each other using a reflection and a translation are:
- Triangles L R K and A R Q (reflected across point R).
- Triangles L P K and Q R A (reflected across a line and then translated).
The third and fourth scenarios do not meet the requirements of a reflection followed by a translation.