Similar triangles may or may not have equal side lengths
Triangles ABC and DEF are similar by SSS theorem
<h3>How to determine the similarity statement</h3>
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
![d = \sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2}](https://tex.z-dn.net/?f=d%20%3D%20%5Csqrt%7B%28x_2%20-x_1%29%5E2%20%2B%20%28y_2%20-y_1%29%5E2%7D)
For triangle ABC, we have:
![AB = \sqrt{(0 -4)^2 + (0 -0)^2} = 4](https://tex.z-dn.net/?f=AB%20%3D%20%5Csqrt%7B%280%20-4%29%5E2%20%2B%20%280%20-0%29%5E2%7D%20%3D%204)
![BC = \sqrt{(4 -0)^2 + (0 -2)^2} = 2\sqrt 5](https://tex.z-dn.net/?f=BC%20%3D%20%5Csqrt%7B%284%20-0%29%5E2%20%2B%20%280%20-2%29%5E2%7D%20%3D%202%5Csqrt%205)
![CA = \sqrt{(0 -0)^2 + (0 -2)^2} = 2](https://tex.z-dn.net/?f=CA%20%3D%20%5Csqrt%7B%280%20-0%29%5E2%20%2B%20%280%20-2%29%5E2%7D%20%3D%202)
For triangle DEF, we have:
![DE = \sqrt{(0 -2)^2 + (0 -0)^2} = 2](https://tex.z-dn.net/?f=DE%20%3D%20%5Csqrt%7B%280%20-2%29%5E2%20%2B%20%280%20-0%29%5E2%7D%20%3D%202)
![EF = \sqrt{(2 -0)^2 + (0 -1)^2} = \sqrt 5](https://tex.z-dn.net/?f=EF%20%3D%20%5Csqrt%7B%282%20-0%29%5E2%20%2B%20%280%20-1%29%5E2%7D%20%3D%20%5Csqrt%205)
![FD = \sqrt{(0 -0)^2 + (1 -0)^2} = 1](https://tex.z-dn.net/?f=FD%20%3D%20%5Csqrt%7B%280%20-0%29%5E2%20%2B%20%281%20-0%29%5E2%7D%20%3D%201)
Divide corresponding sides of the triangles to calculate the scale factor k
![k = \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7BAB%7D%7BDE%7D%20%3D%20%5Cfrac%7BBC%7D%7BEF%7D%20%3D%20%5Cfrac%7BCA%7D%7BFD%7D)
This gives
![k = \frac{4}{2} = \frac{2\sqrt 5}{\sqrt 5} = \frac{2}{1}](https://tex.z-dn.net/?f=k%20%3D%20%5Cfrac%7B4%7D%7B2%7D%20%3D%20%5Cfrac%7B2%5Csqrt%205%7D%7B%5Csqrt%205%7D%20%3D%20%5Cfrac%7B2%7D%7B1%7D)
Evaluate the quotients
![k = 2 = 2 = 2](https://tex.z-dn.net/?f=k%20%3D%202%20%3D%202%20%3D%202)
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:
brainly.com/question/14285697