Question

triangle pqr was dilated according to the rule ( d_{o,2}(x,y)\to(2x,2y) ) to create similar triangle ( pqq ). which statements are true? select two options. ( angle r ) corresponds to ( angle pqq ). ( angle pqr ) corresponds to ( angle qpq ). segment ( qq ) is parallel to segment ( pp ). side ( rq ) corresponds to side ( qq ). ( \triangle pqrcong\triangle pqq )

Explanation:

Step1: Understand dilation properties

Dilation is a transformation that changes the size of a figure but not its shape. Corresponding angles of similar triangles (formed by dilation) are equal.

Step2: Analyze each option

• For \(\angle R\) and \(\angle P'QQ'\):

Since \(\triangle PQR\sim\triangle P'Q'Q\) (by dilation rule \(D_{O,2}(x,y)=(2x,2y)\)), \(\angle R\) and \(\angle P'QQ'\) are corresponding angles. In similar triangles, corresponding angles are congruent.

• For \(\angle PQR\) and \(\angle QPQ'\):

\(\angle PQR\) is an angle of \(\triangle PQR\). \(\angle QPQ'\) is not an angle of \(\triangle P'Q'Q\) in a corresponding position. So they are not corresponding angles.

• For \(QQ'\) and \(PP'\):

Let \(P=(x_1,y_1)\), then \(P'=(2x_1,2y_1)\) and \(Q=(x_2,y_2)\), then \(Q'=(2x_2,2y_2)\). The slope of \(PP'\) is \(m_{PP'}=\frac{2y_1 - y_1}{2x_1 - x_1}=\frac{y_1}{x_1}\) (if \(x_1
eq0\)), and the slope of \(QQ'\) is \(m_{QQ'}=\frac{2y_2 - y_2}{2x_2 - x_2}=\frac{y_2}{x_2}\) (if \(x_2
eq0\)). But from the dilation center \(O\), \(PP'\) and \(QQ'\) are not parallel.

• For \(RQ\) and \(QQ'\):

In \(\triangle PQR\) and \(\triangle P'Q'Q\), \(RQ\) corresponds to \(Q'Q\) (by the order of vertices in similar triangles \(\triangle PQR\sim\triangle P'Q'Q\)).

• For \(\triangle PQR\) and \(\triangle P'Q'Q\):

Since it is a dilation (\(k = 2\)), \(\triangle PQR\sim\triangle P'Q'Q\) (similar, not congruent as size is changed)

Answer:

\(\angle R\) corresponds to \(\angle P'QQ'\); Side \(RQ\) corresponds to side \(QQ'\)