Question

identify whether each pair is not similar, similar and has a scale factor of 1/2, similar and has a scale factor of 4, similar and has a center of dilation at the origin, or similar but does not have a center of dilation at the origin.
figure 1 : △abc
figure 2 : △def

Explanation:

Step1: Find side - lengths of triangles

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the side - lengths of $\triangle ABC$ and $\triangle DEF$. For example, if $A(x_1,y_1)$ and $B(x_2,y_2)$ are two vertices of $\triangle ABC$, then the length of side $AB$ is $\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Do this for all sides of both triangles.

Step2: Calculate the ratios of corresponding sides

Let the side - lengths of $\triangle ABC$ be $a,b,c$ and of $\triangle DEF$ be $a',b',c'$. Calculate the ratios $\frac{a'}{a},\frac{b'}{b},\frac{c'}{c}$. If all the ratios are equal, the triangles are similar. If the ratio is $\frac{1}{2}$, the scale factor is $\frac{1}{2}$; if the ratio is 4, the scale factor is 4.

Step3: Check for center of dilation

If the lines connecting corresponding vertices of the two similar triangles intersect at the origin, the center of dilation is at the origin. If they intersect at a non - origin point, the center of dilation is not at the origin.

Answer:

Without specific coordinates of the vertices of $\triangle ABC$ and $\triangle DEF$ calculated, we cannot give a definite answer. But the general steps to determine are as above.