Respuesta :
Answer:
The vertices of the quadrilateral after a dilation with scale factor [tex]\frac{2}{3}[/tex] are [tex]J'(x,y) = (0,0)[/tex], [tex]K'(x,y) = (0,4)[/tex], [tex]L'(x,y) = (6, 8)[/tex] and [tex]M'(x,y) = (8,0)[/tex].
Step-by-step explanation:
Let suppose that center of dilation is located at origin, that is [tex]O(x,y) = (0,0)[/tex]. Vectorially speaking, dilation is described by following expression:
[tex]D'(x,y) = O(x,y) + r\cdot [D(x,y)-O(x,y)][/tex] (1)
Where:
[tex]O(x,y)[/tex] - Center of dilation.
[tex]D(x,y)[/tex] - Original point.
[tex]D'(x,y)[/tex] - Dilated point.
[tex]r[/tex] - Dilation factor.
If we know that [tex]O(x,y) = (0,0)[/tex], [tex]r = \frac{2}{3}[/tex], [tex]J(x,y) = (0,0)[/tex], [tex]K(x,y) = (0,6)[/tex], [tex]L(x,y) = (9,12)[/tex] and [tex]M(x,y) = (12,0)[/tex], then the dilated points are, respectively:
[tex]J'(x,y) = O(x,y) + r\cdot [J(x,y)-O(x,y)][/tex]
[tex]J'(x,y) = (0,0) +\frac{2}{3}\cdot [(0,0)-(0,0)][/tex]
[tex]J'(x,y) = (0,0)[/tex]
[tex]K'(x,y) = O(x,y) + r\cdot [K(x,y)-O(x,y)][/tex]
[tex]K'(x,y) = (0,0)+\frac{2}{3}\cdot [(0,6)-(0,0)][/tex]
[tex]K'(x,y) = (0,4)[/tex]
[tex]L'(x,y) = O(x,y) +r\cdot [L(x,y)-O(x,y)][/tex]
[tex]L'(x,y) = (0,0) + \frac{2}{3}\cdot [(9,12)-(0,0)][/tex]
[tex]L'(x,y) = (6, 8)[/tex]
[tex]M'(x,y) = O(x,y) +r\cdot [M(x,y)-O(x,y)][/tex]
[tex]M'(x,y) = (0,0) +\frac{2}{3}\cdot [(12,0)-(0,0)][/tex]
[tex]M'(x,y) = (8,0)[/tex]
The vertices of the quadrilateral after a dilation with scale factor [tex]\frac{2}{3}[/tex] are [tex]J'(x,y) = (0,0)[/tex], [tex]K'(x,y) = (0,4)[/tex], [tex]L'(x,y) = (6, 8)[/tex] and [tex]M'(x,y) = (8,0)[/tex].
Option B will be the correct option.
Dilation of a point about the origin by a scale factor 'k',
- If a point (a, b) is dilated by a scale factor 'k' about the origin, coordinates of the image point will be,
(a, b) → (ka, kb)
Following the rule for dilation,
If a quadrilateral J(0, 0), K(0, 6), L(9, 12) and M(12, 0) is dilated by a scale factor [tex]\frac{2}{3}[/tex], coordinates of the image points will be,
[tex]J(0,0)\rightarrow J'(0, 0)[/tex]
[tex]K(0,4)\rightarrow K'(0, 4)[/tex]
[tex]L(9,12)\rightarrow L'(6,8)[/tex]
[tex]M(12,0)\rightarrow M'(8,0)[/tex]
Therefore, Option B will be the correct option.
Learn more about the rule of dilation here,
https://brainly.com/question/22151723?referrer=searchResults
