C. They are similar by the definition of similarity in terms of a dilation.
The conclusion about the triangles is (c) they are similar by the definition of similarity in terms of a dilation.
The coordinates are given as:
[tex]M = (1,3)[/tex]
[tex]N = (4,9)[/tex]
[tex]O = (7,3)[/tex]
[tex]P =(3,0)[/tex]
[tex]Q = (4,2)[/tex]
[tex]R = (5,0)[/tex]
Start by calculating the lengths of the triangles using the following distance formula
[tex]d = \sqrt{(x_1-x_2)^2 + (y_1 -y_2)^2}[/tex]
So, we have:
[tex]MN = \sqrt{(1 -4)^2 + (3 - 9)^2}[/tex]
[tex]MN = \sqrt{45}[/tex]
[tex]MO = \sqrt{(1 -7)^2 + (3 - 3)^2}[/tex]
[tex]MO = 6[/tex]
[tex]NO = \sqrt{(4 -7)^2 + (9 - 3)^2}[/tex]
[tex]NO = \sqrt{45}[/tex]
[tex]PQ = \sqrt{(3 -4)^2 + (0 - 2)^2}[/tex]
[tex]PQ = \sqrt{5}[/tex]
[tex]PR = \sqrt{(3 -5)^2 + (0 - 0)^2}[/tex]
[tex]PR = 2[/tex]
[tex]QR = \sqrt{(4 -5)^2 + (2 - 0)^2}[/tex]
[tex]QR = \sqrt{5}[/tex]
So, we have:
[tex]MN = \sqrt{45}[/tex] [tex]PQ = \sqrt{5}[/tex]
[tex]MO = 6[/tex] [tex]PR = 2[/tex]
[tex]NO = \sqrt{45}[/tex] [tex]QR = \sqrt{5}[/tex]
The above means that the triangles are not congruent.
So, we determine if they are similar by dividing the corresponding sides
[tex]k = \frac{MN}{PQ} = \sqrt{\frac{45}{5}} = 3[/tex]
[tex]k = \frac{MO}{PR} = \frac{6}{2} = 3[/tex]
[tex]k = \frac{NO}{QR} = \sqrt{\frac{45}{5}} = 3[/tex]
The scale factor is 3.
Hence, the triangles are similar
Read more about similar triangles at:
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