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Similar triangles may or may not have equal side lengths
Triangles ABC and DEF are similar by SSS theorem
The coordinates of the two triangles are:
A (0,0), B (4,0), C (0,2), D (0,0), E (2,0), and F (0,1)
Calculate the lengths of both triangles using the following distance formula
[tex]d = \sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2}[/tex]
For triangle ABC, we have:
[tex]AB = \sqrt{(0 -4)^2 + (0 -0)^2} = 4[/tex]
[tex]BC = \sqrt{(4 -0)^2 + (0 -2)^2} = 2\sqrt 5[/tex]
[tex]CA = \sqrt{(0 -0)^2 + (0 -2)^2} = 2[/tex]
For triangle DEF, we have:
[tex]DE = \sqrt{(0 -2)^2 + (0 -0)^2} = 2[/tex]
[tex]EF = \sqrt{(2 -0)^2 + (0 -1)^2} = \sqrt 5[/tex]
[tex]FD = \sqrt{(0 -0)^2 + (1 -0)^2} = 1[/tex]
Divide corresponding sides of the triangles to calculate the scale factor k
[tex]k = \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}[/tex]
This gives
[tex]k = \frac{4}{2} = \frac{2\sqrt 5}{\sqrt 5} = \frac{2}{1}[/tex]
Evaluate the quotients
[tex]k = 2 = 2 = 2[/tex]
Since the quotients are equal, then the triangles are similar
Hence, triangles ABC and DEF are similar by SSS theorem
Read more about similar triangles at:
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