Answer:
Scale factor is [tex]\frac{1}{4}[/tex].
Step-by-step explanation:
Given the coordinates of [tex]\triangle PQR[/tex] as:
[tex]P(4, 0)\\Q(0, -4)\\R(-8, -4)[/tex]
And the coordinates of [tex]\triangle P' Q' R'[/tex] as:
[tex]P'(1, 0)\\Q'(0, -1)\\R'(-2, -1)[/tex]
To find:
The scaling factor of the dilation to transform the [tex]\triangle PQR[/tex] to [tex]\triangle P' Q' R'[/tex].
Solution:
First of all, let us find the distance between the vertices i.e. the sides of the triangle.
Distance formula:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Where [tex](x_1,y_1), (x_2,y_2)[/tex] are the coordinates of two points between which the distance is to be calculated.
[tex]PQ = \sqrt{4^2+4^2} = 4\sqrt2[/tex]
[tex]QR = \sqrt{8^2+0^2} = 8[/tex]
[tex]PR = \sqrt{4^2+12^2} = 4\sqrt{10}[/tex]
Now, let us find the sides of the [tex]\triangle P' Q' R'[/tex]:
[tex]P' Q' = \sqrt{1^2+1^2} = \sqrt{2}[/tex]
[tex]Q' R' = \sqrt{2^2+0^2} = 2[/tex]
[tex]P' R' = \sqrt{3^2+1^2} = \sqrt{10}[/tex]
We can clearly see that, the sides of [tex]\triangle PQR[/tex] are four times the corresponding sides of [tex]\triangle P' Q' R'[/tex].
Therefore, the scaling factor is [tex]\frac{1}{4}[/tex].
Please refer to the attache image in the answer area.
