Question

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

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 \(C'Q=\frac{1}{2}CQ\) (dilation formula: if a point \(X\) is dilated about a center \(O\) with scale factor \(k\), \(OX'=kOX\)). Then \(CQ = 2\times C'Q\). So \(CQ=6\).

Step3: Find \(B'D'\)

Since the dilation rule is \(D_{Q,\frac{1}{2}}\), and \(BD = 22\), then \(B'D'=\frac{1}{2}BD\) (dilation formula: if a point \(X\) is dilated about a center \(O\) with scale factor \(k\), \(OX'=kOX\)). So \(B'D' = 11\).

Answer:

\(m\angle C'B'D' = 95^{\circ}\), \(CQ=6\), \(B'D'=11\)