![Transformation: Translation Transformation: Translation](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjs-7U6aWdHsdOuZsBFc_fR4g6_rHK9sz8206Qmqn6D-1PN4chvOAied37x8xR5YbnjxLQg3sk5GIBJMXX0p0PwXHR6Ypi81WTLsji6ueHa2erhVc42qYyN0YmBXxQfp9XjtSicvBKYG2k/s16000/Translation.png)
The Translation is a transformation in which
each point of the given object is displaced through definite distance and
direction. The displacement is defined by a translation vector (a, b).
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![Translation: Examples Translation: Examples](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpp1pc7wXaEzcJZOzdAhjI-uef5ReVId1bTB8S42Fc5_wUMuSpCY8RJytTWpyCcCIw8LYRuG2ZwUiz0u4MlgfMIQ49bbR8Vav6mXWfALvr92LVHIE_3A2KTsVhtSdG4sj6MUIH90X5qrYV/s16000/Translation+Examples.png)
The following are the properties of translation:
1.
The object and the image
under the translation are congruent.
2.
The lines joining any point
of the object with its corresponding image are parallel and equal.
Example 1: Find the image of ΔABC under the translation through translation
vector PQ.
![Example 1: Question: Find the image of ΔABC under the translation through translation vector PQ.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiks_r3xUhlYnX7Swi-BxjOnfZkp2wip5wE4UOiVejAqtTjLSAMFDdq3MDltx-3yg8mafIHXE7K-fKgG4SOrxqWCxgwgz5dsxudggCciA6L6338zy_wRoRyzqtvKgZTH351ZqcxJG1-4knG/s16000/Example+1+question.png)
Solution:
To get the image of ΔABC under the translation vector PQ draw
the lines through A, B, and C which are parallel and equal to PQ in the same
direction to get the images of A, B, and C and join them to get ΔA’B’C’.
![Example 1: Solution: Image of ΔABC under the translation through translation vector PQ.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaiwQVAO0leY5yrEUpKLC54f6C3H30zJUonG27ASuMCqMRweMDfAORQ2MOEqt7iCH2_MkUZdVn-pJOeO0_ztIAOJ2veg0hMtl3L_VvFOOZp0m0B2oqNH2ltjT4fCR5jAlQjO24vyJcMlWI/s16000/Example+1+solution.png)
Translation Using Co-ordinates
The image of the geometrical figures under the translation through certain translation vector can be obtained with the help of co-ordinates.
Translation Through Translation Vector T = (a, b)
Let A(2, 1) be a point and T = (2, 3) be the translation
vector. To translate A(2, 1) through translation vector T = (2, 3), it has to
move 2 units right and then 3 units up which is the point (4, 4)
![Translation Through Translation Vector T = (2, 3). Translation Through Translation Vector T = (2, 3).](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeDqYmXQ6ihpzIttQRa-veT9iNbKA0m9c0XMDaO2FPHNta_lvgMK160OfHWKSv2MHlHuecN8YNSMJylSehorj0pUeLX5KbghVmewTMymtOW0fl9WgchqLgzoSybOGd5Xj16Mv49aKRecIu/s16000/Translation+Through+Translation+Vector+%2528a%252C+b%2529.png)
Hence, A’(4, 4) is the image of the point A(2, 1) under the translation vector T = AA’ = (2, 3).
![i.e. A(2, 1) → A’(4, 4)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUc0g4NztmI2kFt3xYAQQaOvedRrM_wNeA8P-6xsOzlwnkA2L9OYPoUsp_7ijZkHzXgh8DiKtv6y-wP3nL-fQBbDvjRIFEEjvXCA2Y-cWKtUduuQJ4Dxog_TMKUePP3AiX3af45nO7m2aI/s16000/Image+of+A%25282%252C+1%2529+under+the+translation+by+translation+vector+%25282%252C+3%2529.png)
Let us see the following table of some other points on the same graph and their corresponding images under the translation through translation vector T = (2, 3).
![Table Table of points and their corresponding images under the translation through translation vector T = (2, 3).](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnefPWJ8TA9x5Exa6msLlr3u15-NmNrWgcLotD5U-iNNqhw735QEncCSYpshzNLZk_fM5WWu9uLLhdcEfEY5DzUm7zrxAnGFsC79CMbY_qDAq7SWW6amSEwUmUjlPWJy30JGijBsCiSEN2/s16000/Table+of+points+under+the+Translatiuon+Through+Vector+%25282%252C+3%2529.png)
From the above table, we can see that the image of any
point under the translation through translation vector (a, b) is obtained by
adding a to x-coordinate and b to y-coordinate of the given point.
![Formula Formula of Translation Through Translation Vector T = (a, b).](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-NDGhoo_SIEp4UEY4a1DEEv_yTGGL90R8hoOa-W1Ll_-0art3Nowt4XC7c-WdqaWbqM-LNYme4JoKBwwIv7IGaxrJAwh02_ZlDUhODRicdX0dyqbCvj7IR9wiPmg0sJP2OLgAXdHMNA2R/s16000/Formula+of+Translation+Through+Translation+Vector+%2528a%252C+b%2529.png)
Worked Out Examples
Example 2: If A(-4, -4), B(-2, -1) and C(-1, -5) are the vertices of a
triangle ABC. Find the co-ordinates of the image of ΔABC under the translation
T =(6, 5). Draw ΔABC and its image on the same graph paper.
Solution:
As A(-4, -4), B(-2, -1) and C(-1, -5) are the vertices of ΔABC, the co-ordinates of the vertices of image of ΔABC can be obtained by using the formula as below:
![Example 2: Translation of points by using formula.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv61Awdpav9pwWDhcNfPlTfuLnNw7jrNB8XaBN6NUxDPAF039eLOm5ZOnkifAMBCVaDvjLaDaTrTTUPms3S2uHHpJPcCrhCmg6cKgTSBtGhRvU2tP2WTmcPb1OBf6ey4xtZaZ-t7pOzLXB/s16000/Example+2+translation+of+points+by+using+formula.png)
Drawing ΔABC and ΔA’B’C’ on the same graph paper we have figure
as shown.
![Graph Example 2: Graph.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5SuVVwcQM_ZInEXWqIqacNwugIATLK5PtGgtGmcD9oYXATzWAOUMmCxX2zuBCMb_E9eMwYo9v9oR-bTpj39jt4UpNa559cMcPd0BK4gRag-1LT24Qb2XAKBOXPedQGBQNlV_T5szHUA0f/s16000/Example+2+graph.png)
Example 3: A(-4, 2), B(0, 1) and C(-2, -3) are the vertices of ΔABC. If A’(-1,
6) be the image under the translation of vertex A. Find the images B’ and
C’ of B and C under the same translation.
Solution:
Here, as the image of A(-4, 2) under the translation is A’(-1,
6),
![The translation vector T = ("AA'" ) ⃗ = ("OA'" ) ⃗ - ("OA" ) ⃗ = (■("-1" @"6" )) - (■("-4" @"2" )) = (■("-1+4" @"6-2" )) = (■("3" @"4" )) = (3, 4)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS81zAVXhZhSwAOyg0yu95hpfp3GQ7aMzsbza0xOI86LLkSYp9KQhuA1D7ofE_72319LW9_PdFd9344VyXrwP__iaChiQUaGyfbE0TCZhAsOLcNNuYF2Eq_NX_P8Rj-aTaiN10q1Iqvvgb/s16000/Example+3+solution.png)
Now, under the same translation, the image of B(0, 1) is B’(0+3,
1+4) i.e. B’(3, 5) and the image of C(-2, -3) is C’(-2+3, -3+4) i.e. C’(1, 1).
Example 4: Translate a point A(2, 7) under translation T1 = (4, -3)
to the point A’. Translate A’ to A’’ under another translation T2 =(2,
4). Find the translation vector which translates A to A’’.
Solution:
![Here, the image of A(2, 7) under the translation T1 = (4, -3) is A’(2+4, 7-3) i.e. A’(6, 4). Again the image of A’(6, 4) under the translation T2 = (2, 4) is A’’(6+2, 4+4) i.e. A’’(8, 8)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEray2bwcN-ymadoGPh9PZXqjLXfs2z2522jk-KrjSSFTo5M2AyrWldYJWmwqroXHS_93Joss5b3lSwVVeQYO1-1Oo2s0wIsX79RHwSqZDnscd1Tv7wdeUnU2TAReJtK14fEkZX2DGpR21/s16000/Example+4+solution+part+1.png)
Now, let T be the translation vector which translates A(2, 7) to
A’’(8, 8).
![Hence, T = ("AA''" ) ⃗ = ("OA''" ) ⃗ - ("OA" ) ⃗ = (■("8" @"8" )) - (■("2" @"7" )) = (■("8-2" @"8-7" )) = (■("6" @"1" )) = (6, 1)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjt7A2ACEbIFw9WC8xYBQFm0fabf0-5yU9H4onxVr_xURfEhoqOMxcKz4dV4AtqAnsCv4tJgkqYgVLsTQipRPCd9JWxqB5dTKi0j__3JnG5ZsbhEzUWk-CXmTB7PLavK7dko1ZqE2CB76hU/s16000/Example+4+solution.png)
Example 5: Translate the point M(4, -5) by the translation vector T = (-3, 4)
and find the image point M’ and write down the translation vector which maps M’
to M.
Solution:
![We have, P(x, y) → P’(x+a, y+b) Hence, M(4, -5) → M’(1, -1)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgW04eGODPIaE2ylk4J8ZUt25HCBHv8y6RpIIDa6sbd88r9LA6nzC6p2ISBoMc5ObnHnyn7Mq84cqQa_s6nVlBa6N1WVi7Hnvfn3CIZRlqGZwaSgKnjV2HLTdzoVLkjHxxW5Z5yttWnrLks/s16000/Example+5+solution+part+1.png)
Let T’ = (a, b) be the translation vector which maps M’(1, -1)
back to M(4, -5). Then,
![M’(1, -1) → M(1+a, -1+b) = M(4, -5)](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_StbCPTWmPiGuLWbxfK4XFEeaEZjfnVU1q5Tf_va4tCuuvJ0bBR_UBqvAuhVidF6xoFu7asAaYbdxvgjKelaFq-l8njEkWfOvQ0JsZZjovzWx7WZP9KjCl5oUcvIoKe7fAyS36PNetT6Y/s16000/Example+5+solution+part+2.png)
Therefore,
1+a = 4
or, a = 4 – 1
or, a = 3
And,
-1+b = -5
or, b = -5 + 1
or, b = -4
∴ The required translation vector is (3, -4) which is called the inverse of T and written as T-1.
i.e. If T = (a, b) then
T-1 = (-a, -b).
Am vary happy for your lesson
ReplyDeleteGreat and straight forward lesson. Making Translation much easier
ReplyDeleteThank you.
A point p is translated by vector (3,4)to point p(6,5).find the coordinates of p
ReplyDeleteCoordinates of P = (6-3, 5-4) = (3, 1)
Delete