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 defsim\triangle def )

Explanation:

Step1: Analyze angle correspondence

In a dilation, corresponding angles are congruent. So, \(\angle F\) corresponds to \(\angle F'\) (because dilation is a similarity transformation and preserves angle measures).

Step2: Check parallelism

For dilation with center \(O\), segments like \(EE'\) and \(FF'\) are not parallel. The lines \(EE'\) and \(FF'\) pass through the center of dilation \(O\) (by the property of dilation: if \(P(x,y)\) is dilated to \(P'(kx,ky)\) with center at the origin \(O(0,0)\), then the line \(PP'\) passes through \(O\)).

Step3: Examine distance from origin

If \(D(x,y)\) is dilated by a scale factor \(k = \frac{1}{3}\) with center at the origin \(O(0,0)\) to \(D'(\frac{1}{3}x,\frac{1}{3}y)\), the distance formula \(d=\sqrt{x^{2}+y^{2}}\) for \(D\) and \(d'=\sqrt{(\frac{1}{3}x)^{2}+(\frac{1}{3}y)^{2}}=\frac{1}{3}\sqrt{x^{2}+y^{2}}\). So, the distance from point \(D'\) to the origin is \(\frac{1}{3}\) the distance of point \(D\) to the origin.

Step4: Analyze angle measure

Since dilation is a similarity transformation, \(\angle E=\angle E'\) (angle - measure is preserved in similarity transformations, \(m\angle E'=m\angle E\)).

Step5: Check similarity

Since dilation is a similarity transformation, \(\triangle DEF\sim\triangle D'E'F'\) (by the definition of dilation: a dilation with scale factor \(k
eq0\) is a similarity transformation, and if \(\triangle DEF\) is dilated to \(\triangle D'E'F'\), then \(\triangle DEF\sim\triangle D'E'F'\)).

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, \(\triangle DEF\sim\triangle D'E'F'\)