Question

triangle def was dilated according to the rule $d_{o,\frac{1}{3}}(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)$ to create similar triangle $def$. which statements are true? select three options. $\angle f$ corresponds to $\angle f$. segment $ee$ is parallel to segment $ff$. the distance from point $d$ to the origin is $\frac{1}{3}$ the distance of point $d$ to the origin. the measure of $\angle e$ is $\frac{1}{3}$ the measure of $\angle e$. $\triangle def\sim\triangle def$

Explanation:

Step1: Properties of Dilation

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding angles of similar figures (pre - image and image after dilation) are equal. So, \(\angle F\) corresponds to \(\angle F'\) because dilation preserves angle measures. Also, if \(\triangle DEF\) is dilated to \(\triangle D'E'F'\) with a scale factor \(k = \frac{1}{3}\), then \(\triangle DEF\sim\triangle D'E'F'\) (by the definition of dilation, which creates similar figures).

Step2: Distance in Dilation

The scale factor of the dilation \(D_{O,\frac{1}{3}}(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\) means that if we consider a point \(P(x,y)\) and its image \(P'( \frac{1}{3}x,\frac{1}{3}y)\), the distance from the origin \(O(0,0)\) to \(P\) is \(d=\sqrt{x^{2}+y^{2}}\) and the distance from the origin \(O(0,0)\) to \(P'\) is \(d'=\sqrt{(\frac{1}{3}x)^{2}+(\frac{1}{3}y)^{2}}=\frac{1}{3}\sqrt{x^{2}+y^{2}}\). So, the distance from point \(D'\) to the origin is \(\frac{1}{3}\) the distance of point \(D\) to the origin.

Step3: Analyze False Statements

• For the statement about parallel segments: If we consider the lines \(EE'\) and \(FF'\), since \(E(x_E,y_E)\) and \(E'(\frac{1}{3}x_E,\frac{1}{3}y_E)\), \(F(x_F,y_F)\) and \(F'(\frac{1}{3}x_F,\frac{1}{3}y_F)\). The slope of \(EE'\) is \(m_{EE'}=\frac{\frac{1}{3}y_E - y_E}{\frac{1}{3}x_E - x_E}=\frac{y_E}{x_E}\) (assuming \(x_E

eq0\)), and the slope of \(FF'\) is \(m_{FF'}=\frac{\frac{1}{3}y_F - y_F}{\frac{1}{3}x_F - x_F}=\frac{y_F}{x_F}\). Since \(x_E
eq x_F\) and \(y_E
eq y_F\) in general (from the graph), \(m_{EE'}
eq m_{FF'}\), so \(EE'\) is not parallel to \(FF'\).

• For the angle - measure statement: Since dilation preserves angle measures, \(m\angle E'=m\angle E\) (not \(m\angle E'=\frac{1}{3}m\angle E\))

Answer:

\(\angle F\) corresponds to \(\angle F'\), The distance from point \(D'\) to the origin is \(\frac{1}{3}\) the distance of point \(D\) to the origin, \(\triangle DEF\sim\triangle D'E'F'\)