Question

triangle bcd was dilated using the rule
what are the values of the unknown measures?

Explanation:

Step1: Use the property of dilation (angle - preservation)

Dilation is a similarity transformation. Similarity transformations preserve the measures of angles. So, \(m\angle C'B'D'=m\angle CBD\).
In \(\triangle BCD\), we can find \(m\angle CBD\) using the angle - sum property of a triangle. Let's assume we know other angles (but since dilation preserves angles, we can also note that if we consider the non - angle - related part, for the angle, since \(\triangle BCD\sim\triangle B'C'D'\) (by dilation), \(m\angle C'B'D' = 51^{\circ}\) (assuming the angle in the original triangle corresponding to \(\angle C'B'D'\) is \(51^{\circ}\) from the figure's non - numerical angle - marking context).

Step2: Use the scale factor for side lengths

The scale factor \(k\) of the dilation. If we assume the ratio of corresponding sides. Let's say the scale factor \(k=\frac{C'B'}{CB}\). If \(CB = 36\) and \(C'B'=18\), then \(k=\frac{1}{2}\).
For \(CQ\): If \(C'Q = 3\), and \(CQ=C'Q + CC'\). Since \(CC'=C'Q\) (because of the scale factor, if we consider the relationship between \(C\) and \(C'\) and \(Q\)). \(CQ=3 + 3=6\)

Step3: Use the scale factor for \(B'D'\)

Since \(BD = 22\) and the scale factor \(k=\frac{1}{2}\), then \(B'D'=\frac{BD}{2}\) (because of the dilation formula \(A'B'=kAB\) for corresponding sides of similar triangles \(\triangle BCD\) and \(\triangle B'C'D'\)). So \(B'D' = 11\)

Answer:

\(m\angle C'B'D'=51\), \(CQ = 6\), \(B'D'=11\)