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Sagot :
Let's reflect the coordinates step-by-step according to the given instructions.
### Step 1: Reflecting Over [tex]\( y = -1 \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] over the line [tex]\(y = -1\)[/tex], we can use the formula:
[tex]\[ y' = -1 - (y - (-1)) \][/tex]
This simplifies to:
[tex]\[ y' = -1 - y + 1 = -y - 1 + 1 = -y - 2 + 2 - 1 = -2y -1 + 1 = -y + 1\][/tex]
Let's apply this to the coordinates of the points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
1. Point [tex]\(A(8, 8)\)[/tex]:
[tex]\[ y'_A = -1 - (8 - (-1)) = -1 - (8 + 1) = -1 - 9 = -10 \][/tex]
So, [tex]\( A'(8, -10) \)[/tex].
2. Point [tex]\(B(10, 6)\)[/tex]:
[tex]\[ y'_B = -1 - (6 - (-1)) = -1 - (6 + 1) = -1 - 7 = -8 \][/tex]
So, [tex]\( B'(10, -8) \)[/tex].
3. Point [tex]\(C(2, 2)\)[/tex]:
[tex]\[ y'_C = -1 - (2 - (-1)) = -1 - (2 + 1) = -1 - 3 = -4 \][/tex]
So, [tex]\( C'(2, -4) \)[/tex].
The coordinates after reflection over [tex]\( y = -1 \)[/tex] are:
[tex]\[ \begin{array}{l} A'(8, -10) \\ B'(10, -8) \\ C'(2, -4) \end{array} \][/tex]
### Step 2: Reflecting Over [tex]\( y = -7 \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] over the line [tex]\(y = -7\)[/tex], we can use the formula:
[tex]\[ y'' = -7 - (y' - (-7)) \][/tex]
This simplifies to:
[tex]\[ y'' = -7 - y' + 7 = - y' \][/tex]
Let's apply this to the new coordinates:
1. Point [tex]\(A'(8, -10)\)[/tex]:
[tex]\[ y''_A = -7 - (-10 - (-7)) = -7 - (-10 + 7) = -7 - (-3) = -7+ 3 = -4 \][/tex]
So, [tex]\( A''(8, -4) \)[/tex].
2. Point [tex]\(B'(10, -8)\)[/tex]:
[tex]\[ y''_B = -7 - (-8 - (-7)) = -7 - (-8 + 7) = -7 - (-1) = -7+1 \][/tex]
So, [tex]\( B''(10, -6) \)[/tex].
3. Point [tex]\(C'(2, -4)\)[/tex]:
[tex]\[ y''_C = -7 - (-4 - (-7)) = -7 -(-4+7)= -7 -3 \][/tex]
[tex]\[ y''_C =4+(-7-\sqrt{36}) \][/tex]
So, [tex]\( C''(2, -10) ). The final coordinates after reflection over \( y = -7 \)[/tex] are:
[tex]\[ \begin{array}{l} A''(8, -4) \\ B''(10, -6) \\ C''(2, -10) \end{array} \][/tex]
Thus, the values of [tex]\( y' \)[/tex] for points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are [tex]\(-10\)[/tex], [tex]\(-8\)[/tex], and [tex]\(-4\)[/tex] respectively, and the final reflected coordinates are [tex]\(A''(8, -4)\)[/tex], [tex]\(B''(10, -6)\)[/tex], and [tex]\(C''(2, -10)\)[/tex].
### Step 1: Reflecting Over [tex]\( y = -1 \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] over the line [tex]\(y = -1\)[/tex], we can use the formula:
[tex]\[ y' = -1 - (y - (-1)) \][/tex]
This simplifies to:
[tex]\[ y' = -1 - y + 1 = -y - 1 + 1 = -y - 2 + 2 - 1 = -2y -1 + 1 = -y + 1\][/tex]
Let's apply this to the coordinates of the points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
1. Point [tex]\(A(8, 8)\)[/tex]:
[tex]\[ y'_A = -1 - (8 - (-1)) = -1 - (8 + 1) = -1 - 9 = -10 \][/tex]
So, [tex]\( A'(8, -10) \)[/tex].
2. Point [tex]\(B(10, 6)\)[/tex]:
[tex]\[ y'_B = -1 - (6 - (-1)) = -1 - (6 + 1) = -1 - 7 = -8 \][/tex]
So, [tex]\( B'(10, -8) \)[/tex].
3. Point [tex]\(C(2, 2)\)[/tex]:
[tex]\[ y'_C = -1 - (2 - (-1)) = -1 - (2 + 1) = -1 - 3 = -4 \][/tex]
So, [tex]\( C'(2, -4) \)[/tex].
The coordinates after reflection over [tex]\( y = -1 \)[/tex] are:
[tex]\[ \begin{array}{l} A'(8, -10) \\ B'(10, -8) \\ C'(2, -4) \end{array} \][/tex]
### Step 2: Reflecting Over [tex]\( y = -7 \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] over the line [tex]\(y = -7\)[/tex], we can use the formula:
[tex]\[ y'' = -7 - (y' - (-7)) \][/tex]
This simplifies to:
[tex]\[ y'' = -7 - y' + 7 = - y' \][/tex]
Let's apply this to the new coordinates:
1. Point [tex]\(A'(8, -10)\)[/tex]:
[tex]\[ y''_A = -7 - (-10 - (-7)) = -7 - (-10 + 7) = -7 - (-3) = -7+ 3 = -4 \][/tex]
So, [tex]\( A''(8, -4) \)[/tex].
2. Point [tex]\(B'(10, -8)\)[/tex]:
[tex]\[ y''_B = -7 - (-8 - (-7)) = -7 - (-8 + 7) = -7 - (-1) = -7+1 \][/tex]
So, [tex]\( B''(10, -6) \)[/tex].
3. Point [tex]\(C'(2, -4)\)[/tex]:
[tex]\[ y''_C = -7 - (-4 - (-7)) = -7 -(-4+7)= -7 -3 \][/tex]
[tex]\[ y''_C =4+(-7-\sqrt{36}) \][/tex]
So, [tex]\( C''(2, -10) ). The final coordinates after reflection over \( y = -7 \)[/tex] are:
[tex]\[ \begin{array}{l} A''(8, -4) \\ B''(10, -6) \\ C''(2, -10) \end{array} \][/tex]
Thus, the values of [tex]\( y' \)[/tex] for points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are [tex]\(-10\)[/tex], [tex]\(-8\)[/tex], and [tex]\(-4\)[/tex] respectively, and the final reflected coordinates are [tex]\(A''(8, -4)\)[/tex], [tex]\(B''(10, -6)\)[/tex], and [tex]\(C''(2, -10)\)[/tex].
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