Question

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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Answer 1

(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)
`\(T_1\):`

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