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:

planation:

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 (\(180^{\circ}\)), \(m\angle CBD = 180^{\circ}-34^{\circ}-51^{\circ}=95^{\circ}\).

Step2: Find \(CQ\)

Since \(C'Q = 3\) and \(CC'=C'Q\) (because dilation with center \(Q\) and scale factor \(\frac{1}{2}\), \(CQ = 2\times C'Q\)). So \(CQ=6\).

Step3: Find \(B'D'\)

Given \(BD = 22\), and dilation scale factor \(k=\frac{1}{2}\). Using the formula for dilation \(B'D'=\frac{1}{2}BD\). So \(B'D' = 11\).

Answer:

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