Respuesta :
The translation rule which describes the given translation is [tex]\fbox{\begin\\\math{(x,y)}\rightarrow(x+5,y-3)\\\end{minispace}}[/tex].
Further explanation:
Translation is defined as the shifting of the curve in which each point on the curve is shifted in such a way that the shape and size of the curve remains unchanged but the position of each point is changed by a fixed distance.
In the given figure there are two right angle triangle as [tex]S'T'U'[/tex] and [tex]STU[/tex].
The coordinates of the right angle triangle [tex]S'T'U'[/tex] are [tex](-4,2),(-4,-2)[/tex] and [tex](-1,-2)[/tex] and the coordinates of the right angle triangle [tex]STU[/tex] are [tex](1,5),(1,1)[/tex] and [tex](4,1)[/tex].
Consider that the triangle [tex]STU[/tex] is obtained after the translation of each point on the triangle [tex]S'T'U'[/tex].
The coordinate of the point [tex]S'[/tex] in triangle [tex]S'T'U'[/tex] is [tex](-4,2)[/tex].
Step 1:
Use the translation [tex](x,y)\rightarrow(x-5,y-3)[/tex] for the point [tex]S'[/tex].
[tex]\fbox{\begin\\\(-4,2)\rightarrow(-4-5,2-3)=(-9,-1)\\\end{minispace}}[/tex]
As per the translation [tex](x,y)\rightarrow(x-5,y-3)[/tex] the coordinate of the point [tex]S[/tex] is [tex](-9,-1)[/tex], but in the given figure the coordinate of the point [tex]S[/tex] is [tex](1,5)[/tex].
So, the translation [tex](x,y)\rightarrow(x-5,y-3)[/tex] is incorrect.
Step 2:
Use the translation [tex](x,y)\rightarrow(x-3,y-5)[/tex] for the point [tex]S'[/tex].
[tex]\fbox{\begin\\\(-4,2)\rightarrow(-4-3,2-5)=(-7,-3)\\\end{minispace}}[/tex]
As per the translation [tex](x,y)\rightarrow(x-3,y-5)[/tex] the coordinate of the point [tex]S[/tex] is [tex](-7,-3)[/tex], but in the given figure the coordinate of the point [tex]S[/tex] is [tex](1,5)[/tex].
So, the translation [tex](x,y)\rightarrow(x-3,y-5)[/tex] is incorrect.
Step 3:
Use the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] for the point [tex]S'[/tex].
[tex]\fbox{\begin\\\(-4,2)\rightarrow(-4+5,2+3)=(1,5)\\\end{minispace}}[/tex]
As per the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] the coordinate of the point [tex]S[/tex] is [tex](1,5)[/tex] which is correct as the coordinate for the point [tex]S[/tex] given in the figure is [tex](1,5)[/tex].
Step 4:
Use the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] for the point [tex]T'[/tex].
[tex]\fbox{\begin\\\(-4,-2)\rightarrow(-4+5,-2+3)=(1,1)\\\end{minispace}}[/tex]
As per the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] the coordinate of the point [tex]T[/tex] is [tex](1,1)[/tex] which is correct as the coordinate for the point [tex]T[/tex] given in the figure is [tex](1,1)[/tex].
Step 5:
Use the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] for the point U’.
[tex]\fbox{\begin\\\ (-1,-2)\rightarrow(-1+5,-2+3)=(4,1)\\\end{minispace}}[/tex]
As per the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] the coordinate of the point [tex]T[/tex] is [tex](4,1)[/tex] which is correct as the coordinate for the point [tex]U[/tex] given in the figure is [tex](4,1)[/tex].
As per the above calculation it is concluded that the translation [tex](x,y)\rightarrow(x+5,y+3)[/tex] is correct for the shifting of each point on the triangle [tex]S'T'U'[/tex] to each point on the triangle [tex]STU[/tex].
The translation of each point on the triangle [tex]S'T'U'[/tex] is shown in the image attached below.
Therefore, the correct translation rule for the shifting of the triangle [tex]S'T'U'[/tex] to the triangle [tex]STU[/tex] is [tex] \fbox{\begin\\\math{(x,y)}\rightarrow(x+5,y-3)\\\end{minispace}}[/tex]
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Answer details:
Grade: Middle school
Subject: Mathematics
Chapter: Coordinate geometry
Keywords: Translation, shifting, graph, geometry, coordinate geometry, coordinates, shifting of graph, triangle, right angle triangle, movement of curve, shifting of curve, translation of curve.
Answer:
(x, y)⇒(x-5, y-3)
Step-by-step explanation:
In the original figure we have the following vertices:
T(1, 1)
U(4, 1)
S(1, 5)
And in the translated figure we have the new vertices:
T'(-4, -2)
U'(-1, -2)
S'(-4, 2)
We can observe that from T to T 'the component in x was reduced by 5 and the component in y was reduced by 3:
(1 - 5, 1 - 3) = ( -4, -2) wich is T'
And the same goes for U and S
For U:
(4 - 5, 1 - 3) = (-1, -2) wich is U'
For S:
(1 - 5, 5 - 3) = (-4, 2) wich is S'
thus the rule for the translation is:
(x, y)⇒(x-5, y-3)
