At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine which point maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the properties of reflection across this line. A point [tex]\((a, b)\)[/tex] when reflected across the line [tex]\( y = -x \)[/tex] will map to the point [tex]\((-b, -a)\)[/tex].
For a point to map onto itself when reflected across the line [tex]\( y = -x \)[/tex], the point must satisfy the condition:
[tex]\[ (a, b) = (-b, -a) \][/tex]
This implies that:
[tex]\[ a = -b \][/tex]
[tex]\[ b = -a \][/tex]
Therefore, we see that the only points that satisfy this condition are those where both coordinates are equal in magnitude but opposite in sign. Specifically for our question, since we want the point to remain unchanged:
[tex]\[ a = -a \][/tex]
[tex]\[ b = -b \][/tex]
This further simplifies to:
[tex]\[ a = -a \Rightarrow a = 0 \][/tex]
[tex]\[ b = -b \Rightarrow b = 0 \][/tex]
So, a point that maps onto itself must satisfy [tex]\( x = -y \)[/tex]. Now, let's check each given point:
1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = 0 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[ x = 0 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x = -y \)[/tex], so this point does satisfy the condition.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
So, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
Thus, the correct choice number is:
[tex]\[ \boxed{4} \][/tex]
For a point to map onto itself when reflected across the line [tex]\( y = -x \)[/tex], the point must satisfy the condition:
[tex]\[ (a, b) = (-b, -a) \][/tex]
This implies that:
[tex]\[ a = -b \][/tex]
[tex]\[ b = -a \][/tex]
Therefore, we see that the only points that satisfy this condition are those where both coordinates are equal in magnitude but opposite in sign. Specifically for our question, since we want the point to remain unchanged:
[tex]\[ a = -a \][/tex]
[tex]\[ b = -b \][/tex]
This further simplifies to:
[tex]\[ a = -a \Rightarrow a = 0 \][/tex]
[tex]\[ b = -b \Rightarrow b = 0 \][/tex]
So, a point that maps onto itself must satisfy [tex]\( x = -y \)[/tex]. Now, let's check each given point:
1. For the point [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
2. For the point [tex]\((-4, 0)\)[/tex]:
[tex]\[ x = -4 \][/tex]
[tex]\[ y = 0 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
3. For the point [tex]\((0, -4)\)[/tex]:
[tex]\[ x = 0 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x \neq -y \)[/tex], so this point does not satisfy the condition.
4. For the point [tex]\((4, -4)\)[/tex]:
[tex]\[ x = 4 \][/tex]
[tex]\[ y = -4 \][/tex]
Here [tex]\( x = -y \)[/tex], so this point does satisfy the condition.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
So, the answer is:
[tex]\[ \boxed{(4, -4)} \][/tex]
Thus, the correct choice number is:
[tex]\[ \boxed{4} \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.