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. ( squareangle f ) corresponds to ( angle f ). ( square ) segment ( ee ) is parallel to segment ( ff ). ( square ) the distance from point ( d ) to the origin is ( \frac{1}{3} ) the distance of point ( d ) to the origin. ( square ) the measure of ( angle e ) is ( \frac{1}{3} ) the measure of ( angle e ). ( square\triangle defsim\triangle def )

Explanation:

Step1: Properties of dilation

Dilation is a similarity transformation. Corresponding angles of similar triangles are equal. So, \(\angle F\) corresponds to \(\angle F'\) and \(\triangle DEF\sim\triangle D'E'F'\).

Step2: Distance in dilation

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

Step3: Parallel segments

Segments \(EE'\) and \(FF'\) are not parallel. For angles, \(\angle E'=\angle E\) (not \(\frac{1}{3}\) of \(\angle E\)) because of the property of similar triangles (corresponding angles are equal).

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