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Under the translation T,, the image of the point (5, -1) is (2, 3). Under a translation T the image of the point (-2, 5) is (4,-5). Find the image of point (7, 6) under the following transformations. (a) T₁ (b) T₂

(c) T₁ T₂ (d) T₂T₁ (e) T₁



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1 Answer

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(a)
\(T_1(7, 6) = (4, 10)\)

(b)
\(T_2(7, 6) = (13, -4)\)

(c)
\(T_1T_2(7, 6) = (10, 0)\)

(d)
\(T_2T_1(7, 6) = (10, 0)\)

(e)
\(T_1(7, 6) = (4, 10)\)

Let's denote the translations as
\(T_1\) and
\(T_2\). The general form of a translation is
\(T(x, y) = (x + a, y + b)\), where
\((a, b)\) are the translation parameters.

Given that under
\(T_1\), the image of the point (5, -1) is (2, 3), we can find \
(T_1\) as follows:


\[ T_1(x, y) = (x - 3, y + 4) \]

Similarly, under \(T_2\), the image of the point (-2, 5) is (4, -5), so:


\[ T_2(x, y) = (x + 6, y - 10) \]

Now, let's find the image of the point (7, 6) under the given transformations:

(a)
\(T_1\):


\[ T_1(7, 6) = (7 - 3, 6 + 4) = (4, 10) \]

(b)
\(T_2\):


\[ T_2(7, 6) = (7 + 6, 6 - 10) = (13, -4) \]

(c)
\(T_1T_2\):


\[ T_1(T_2(7, 6)) = T_1(13, -4) = (13 - 3, -4 + 4) = (10, 0) \]

(d)
\(T_2T_1\):


\[ T_2(T_1(7, 6)) = T_2(4, 10) = (4 + 6, 10 - 10) = (10, 0) \]

(e)
\(T_1\):


\[ T_1(7, 6) = (7 - 3, 6 + 4) = (4, 10) \]

User David Castillo
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