Question

1. write a sequence of transformations to take δabc to δdec

response:

Explanation:

Step1: Analyze Angles and Sides

First, find the third angle in $\triangle ABC$: $\angle B = 180^\circ - 60^\circ - 20^\circ = 100^\circ$. In $\triangle DEC$, $\angle D = 60^\circ$, $\angle E = 100^\circ$, so $\angle C$ (in $\triangle DEC$) would be $20^\circ$, matching $\triangle ABC$. Now, check side ratios: $AC$ (let's assume $AC$ length, but $AB = 9$, $DE = 12$; $BC = 6$, $EC$? Wait, scale factor: $\frac{12}{9} = \frac{4}{3}$, $\frac{EC}{BC}$? Wait, maybe first a dilation.

Step2: Dilation

Dilate $\triangle ABC$ with center at $C$ by a scale factor of $\frac{4}{3}$ (since $DE = 12$, $AB = 9$; $12/9 = 4/3$). So $AB$ becomes $9\times\frac{4}{3}=12$, $BC$ becomes $6\times\frac{4}{3}=8$? Wait, no, maybe center at $C$? Wait, then rotation: after dilation, rotate $\triangle ABC$ (dilated) around point $C$ so that $\angle ACB$ aligns with $\angle DCE$. Wait, $\angle ACB = 20^\circ$? No, earlier $\angle A = 60^\circ$, $\angle ACB = 20^\circ$, $\angle B = 100^\circ$. In $\triangle DEC$, $\angle D = 60^\circ$, $\angle E = 100^\circ$, $\angle DCE = 20^\circ$. So first, dilate $\triangle ABC$ with scale factor $\frac{4}{3}$ (since $DE = 12$, $AB = 9$; $12/9 = 4/3$) centered at $C$. Then, rotate the dilated triangle around point $C$ so that side $BC$ (dilated) aligns with $EC$ and $AC$ (dilated) aligns with $DC$. Alternatively, first rotate, then dilate. Wait, let's check angles: $\angle A = 60^\circ$ matches $\angle D = 60^\circ$, $\angle B = 100^\circ$ matches $\angle E = 100^\circ$, $\angle ACB = 20^\circ$ matches $\angle DCE = 20^\circ$. So the sequence: Dilate $\triangle ABC$ with center $C$ by scale factor $\frac{4}{3}$, then rotate around $C$ to map $A$ to $D$ and $B$ to $E$.

Step3: Confirm Transformations

Dilation: scale factor $\frac{4}{3}$ (since $DE = 12$, $AB = 9$; $12/9 = 4/3$) centered at $C$. Then rotation: rotate the dilated triangle around point $C$ so that $\angle A$ (dilated) aligns with $\angle D$, which is $60^\circ$, and $\angle B$ (dilated) aligns with $\angle E$ ($100^\circ$). This maps $\triangle ABC$ to $\triangle DEC$.

Answer:

First, dilate $\triangle ABC$ with center at point $C$ by a scale factor of $\frac{4}{3}$. Then, rotate the dilated triangle around point $C$ so that $\angle ACB$ (of the dilated triangle) aligns with $\angle DCE$, mapping $\triangle ABC$ to $\triangle DEC$. (Alternative: Rotate first, then dilate, but dilation scale factor is $\frac{4}{3}$ as $DE = 12$, $AB = 9$; $12/9 = 4/3$.)