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Sagot :
To determine which point would map onto itself after a reflection across the line [tex]\( y = -x \)[/tex], we need to understand the properties of such a reflection. When reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the coordinates of the point will be swapped and negated. Specifically, the point [tex]\((x, y)\)[/tex] will be mapped to [tex]\((-y, -x)\)[/tex].
For a point to map onto itself under this reflection, the point [tex]\((x, y)\)[/tex] must satisfy the condition that:
[tex]\[ (x, y) = (-y, -x) \][/tex]
This translates to two separate conditions:
1. [tex]\( x = -y \)[/tex]
2. [tex]\( y = -x \)[/tex]
For both conditions to be true simultaneously, we can deduce that:
[tex]\[ x = -y \][/tex]
[tex]\[ y = -x \][/tex]
From the first condition [tex]\( x = -y \)[/tex], substituting back into the second condition, we also get:
[tex]\[ -y = -(-y) \][/tex]
[tex]\[ y = y \][/tex]
The points we are analyzing are:
1. [tex]\( (-4, -4) \)[/tex]
2. [tex]\( (-4, 0) \)[/tex]
3. [tex]\( (0, -4) \)[/tex]
4. [tex]\( (4, -4) \)[/tex]
Now let’s check each point:
1. For [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ -4 = -(-4) \][/tex]
[tex]\[ -4 = 4 \][/tex]
This condition doesn't hold, so [tex]\((-4, -4)\)[/tex] does not map onto itself.
2. For [tex]\( (-4, 0) \)[/tex]:
[tex]\[ x = -4, y = 0 \][/tex]
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ -4 = -0 \][/tex]
[tex]\[ -4 = 0 \][/tex]
This condition doesn't hold, so [tex]\((-4, 0)\)[/tex] does not map onto itself.
3. For [tex]\( (0, -4) \)[/tex]:
[tex]\[ x = 0, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ 0 = -(-4) \][/tex]
[tex]\[ 0 = 4 \][/tex]
This condition doesn't hold, so [tex]\( (0, -4)\)[/tex] does not map onto itself.
4. For [tex]\( (4, -4) \)[/tex]:
[tex]\[ x = 4, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ 4 = -(-4) \][/tex]
[tex]\[ 4 = 4 \][/tex]
This condition holds true, so [tex]\( (4, -4)\)[/tex] maps onto itself.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
For a point to map onto itself under this reflection, the point [tex]\((x, y)\)[/tex] must satisfy the condition that:
[tex]\[ (x, y) = (-y, -x) \][/tex]
This translates to two separate conditions:
1. [tex]\( x = -y \)[/tex]
2. [tex]\( y = -x \)[/tex]
For both conditions to be true simultaneously, we can deduce that:
[tex]\[ x = -y \][/tex]
[tex]\[ y = -x \][/tex]
From the first condition [tex]\( x = -y \)[/tex], substituting back into the second condition, we also get:
[tex]\[ -y = -(-y) \][/tex]
[tex]\[ y = y \][/tex]
The points we are analyzing are:
1. [tex]\( (-4, -4) \)[/tex]
2. [tex]\( (-4, 0) \)[/tex]
3. [tex]\( (0, -4) \)[/tex]
4. [tex]\( (4, -4) \)[/tex]
Now let’s check each point:
1. For [tex]\((-4, -4)\)[/tex]:
[tex]\[ x = -4, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ -4 = -(-4) \][/tex]
[tex]\[ -4 = 4 \][/tex]
This condition doesn't hold, so [tex]\((-4, -4)\)[/tex] does not map onto itself.
2. For [tex]\( (-4, 0) \)[/tex]:
[tex]\[ x = -4, y = 0 \][/tex]
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ -4 = -0 \][/tex]
[tex]\[ -4 = 0 \][/tex]
This condition doesn't hold, so [tex]\((-4, 0)\)[/tex] does not map onto itself.
3. For [tex]\( (0, -4) \)[/tex]:
[tex]\[ x = 0, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ 0 = -(-4) \][/tex]
[tex]\[ 0 = 4 \][/tex]
This condition doesn't hold, so [tex]\( (0, -4)\)[/tex] does not map onto itself.
4. For [tex]\( (4, -4) \)[/tex]:
[tex]\[ x = 4, y = -4 \][/tex]
Substitute [tex]\( y = -4 \)[/tex] into [tex]\( x = -y \)[/tex]:
[tex]\[ 4 = -(-4) \][/tex]
[tex]\[ 4 = 4 \][/tex]
This condition holds true, so [tex]\( (4, -4)\)[/tex] maps onto itself.
Therefore, the point [tex]\((4, -4)\)[/tex] is the one that maps onto itself after a reflection across the line [tex]\( y = -x \)[/tex].
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