Question

calculating measures of a dilation
triangle bcd was dilated using the rule ( d_{q,\frac{1}{2}} ).
what are the values of the unknown measures?
( mangle cbd=)
( cq=)
( bd=)

Explanation:

Step1: Find \(m\angle C'B'D'\)

Dilation preserves angles. So \(m\angle C'B'D'=m\angle CBD\).
In \(\triangle BCD\), using the angle - sum property of a triangle (\(180^{\circ}\) in a triangle), \(\angle CBD = 180^{\circ}-34^{\circ}-51^{\circ}=95^{\circ}\).

Step2: Find \(CQ\)

Since the dilation rule is \(D_{Q,\frac{1}{2}}\), if \(C'Q = 3\), and \(C'Q=\frac{1}{2}CQ\) (by the definition of dilation with scale factor \(k = \frac{1}{2}\)), then \(CQ=2\times C'Q\).
So \(CQ = 6\).

Step3: Find \(B'D'\)

Given \(BD = 22\), and for a dilation with scale factor \(k=\frac{1}{2}\), the length of the image segment \(B'D'=k\times BD\).
So \(B'D'=\frac{1}{2}\times22 = 11\).

Answer:

\(m\angle C'B'D' = 95\), \(CQ=6\), \(B'D' = 11\)