Two triangles are shown on a coordinate grid. ![](http://admin.getprephub.com/media/editor/t14_20230225221318457385.png) Katie shows that the two triangles are similar by performing the following transformations: - First, she rotates triangle $${ABC} \quad 180^{\circ}$$ about point $${A}$$. - Then, she dilates triangle $$A^{/} B^{/} C^{/}$$by a factor of $${k}$$ with a center of dilation at point $$A$$. - Finally, she translates triangle $$\quad \triangle A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} . p$$ units to the right and $$q$$ units down to map onto $$\triangle CDE$$. What is the value of $$k$$ ?

The co-ordinates of $$A, B, C$$ are $$(4,3),(8,3)$$ and $$(8,0)$$ respectively. Distance $$A B=\sqrt{(4-8)^{2}+(3-3)^{2}}=4$$ units The co-ordinates of $$C, D, E$$ are $$(8,0),(0,0)$$ and $$(0,6)$$ respectively. Distance $$C D=\sqrt{(0-8)^{2}+(0-0)^{2}}=8$$ units Scale factor $$(k)=\frac{8}{4}=2$$

Prephub

Two triangles are shown on a coordinate grid.

Katie shows that the two triangles are similar by performing the following transformations: - First, she rotates triangle ${ABC} \quad 180^{\circ}$ about point ${A}$ .

Then, she dilates triangle $A^{/} B^{/} C^{/}$ by a factor of ${k}$ with a center of dilation at point $A$ .
Finally, she translates triangle $\quad \triangle A^{\prime \prime} B^{\prime \prime} C^{\prime \prime} . p$ units to the right and $q$ units down to map onto $\triangle CDE$ . What is the value of $k$ ?

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