Drag the vertices of the given triangle and observe how the coordinates matrix changes accordingly.[br][br]Choose a predefined transformation or create your custom one using the appearing sliders.[br][br]The coordinates of the transformed triangle can be obtained by multiplying the transformation matrix by the given triangle's coordinates matrix.
Given the transformation matrix [math]T=\begin{pmatrix}[br]1 & 2\\[br]2 & 1[br]\end{pmatrix} [/math], that maps [math]\left(x,y\right)\rightarrow\left(x',y'\right)[/math] , write the equations of the transformation, then find the images of the points [math]O=\left(0,0\right),A=\left(1,2\right)[/math] and [math]B=\left(-1,1\right)[/math].[br][br]Use the app above to check your results, by selecting the [i]Custom[/i] option and setting the matrix using the displayed sliders.
Select [i]Dilation[/i] from the list of transformations in the app above.[br]Observe the measures of the areas displayed, and how they change when you drag the slider. [br][br]In particular, check the values obtained when [math]k=\pm1[/math] and [math]k=\pm2[/math].[br][br]What is the relationship between the areas of the given triangle and its image?[br]Does this relationship depend on the dilation ratio [math]k[/math]?
Select [i]Dilation[/i] from the list of transformations in the app above, and set the ratio [math]k=-1[/math].[br][br]Describe the relative position of the given triangle and its image.[br]Observe the transformation matrix.[br][br]Now, without modifying the given triangle, select [i]Rotation[/i], and apply a 180° rotation to the given triangle.[br][br]What do you notice? Can you generalize this?[br]