Respuesta :
Answer:
The dilation on any point of the rectangle is [tex]P'(x,y) = \frac{5}{2}\cdot P(x,y)[/tex].
Step-by-step explanation:
From Linear Algebra, we define the dilation of a point by means of the following definition:
[tex]G'(x,y) = O(x,y) +k\cdot [G(x,y)-O(x,y)][/tex] (1)
Where:
[tex]G(x,y)[/tex] - Coordinates of the point G, dimensionless.
[tex]O(x,y)[/tex] - Center of dilation, dimensionless.
[tex]k[/tex] - Scale factor, dimensionless.
[tex]G'(x,y)[/tex] - Coordinates of the point G', dimensionless.
If we know that [tex]O(x,y) = (0,0)[/tex], [tex]G(x,y) =(2,-2)[/tex] and [tex]G'(x,y) =(5,-5)[/tex], then scale factor is:
[tex](5,-5) = (0,0) +k\cdot [(2,-2)-(0,0)][/tex]
[tex](5,-5) = (2\cdot k, -2\cdot k)[/tex]
[tex]k = \frac{5}{2}[/tex]
The dilation on any point of the rectangle is:
[tex]P'(x,y) = (0,0) + \frac{5}{2}\cdot [P(x,y)-(0,0)][/tex]
[tex]P'(x,y) = \frac{5}{2}\cdot P(x,y)[/tex] (2)
The dilation on any point of the rectangle is [tex]P'(x,y) = \frac{5}{2}\cdot P(x,y)[/tex].
Algebraic representation of the dilation will be T'(x, y) → T(2.5x, 2.5y).
Dilation of a rectangle about a point not the origin:
- Let one of the four vertices of a rectangle is (h, k) is dilated about the origin by the scale factor 'k'.
Rule for the transformation will be,
[tex]D_{(o,k)}(x,y)\rightarrow (kx,ky)[/tex]
Following this rule,
- If one of the vertices of a rectangle is G(2, -2) and it is dilated by a scale factor 'k', image point will be given by,
[tex]D_{(o,k)}(2, -2)\rightarrow (2k, -2k)[/tex]
Since, coordinates of the dilated point are G'(5, -5),
Therefore, (2k, -2k) = (5, -5)
⇒ 2k = 5
⇒ k = 2.5
Hence, algebraic representation defining the dilation of the
triangle will be,
T'(x, y) → T(2.5x, 2.5y)
Algebraic representation of the dilation will be T'(x, y) → T(2.5x, 2.5y).
Learn more about the transformation here,
https://brainly.com/question/16510020?referrer=searchResults
