Question

triangle bcd was dilated using the rule $d_{q, \frac{1}{2}}$. what are the values of the unknown measures? $m\angle c b d = \square ^\circ$, $cq = \square$, $b d = \square$

Explanation:

Step1: Analyze angle preservation in dilation

Dilation preserves angle measures, so \( m\angle C'B'D' = m\angle CBD = 51^\circ \).

Step2: Determine scale factor

The scale factor \( k \) is \( \frac{18}{36}=\frac{1}{2} \), or from \( D'D = 2 \), \( BD = 22 \), but using \( C'C = 3 \), let \( CQ = x \), \( C'Q = x - 3 \). Since \( k=\frac{1}{2} \), \( \frac{C'Q}{CQ}=\frac{1}{2} \), so \( CQ = 2C'Q \), and \( CQ - C'Q = 3 \), \( C'Q = 3 \), \( CQ = 6 \).

Step3: Calculate \( B'D' \)

Using scale factor \( \frac{1}{2} \), \( B'D'=\frac{1}{2}BD=\frac{22}{2}=11 \)? Wait, no, wait: Wait, the small triangle has \( B'C' = 18 \), big is \( 36 \), so scale factor \( \frac{1}{2} \). \( BD = 22 \), so \( B'D' = \frac{22}{2}=11 \)? Wait, but maybe I misread. Wait, the length from \( B \) to \( D \) is 22, so \( B'D' = 22\times\frac{1}{2}=11 \)? Wait, no, the small triangle's \( B'D' \): Wait, the scale factor is \( \frac{18}{36}=\frac{1}{2} \), so all lengths scale by \( \frac{1}{2} \). So \( BD = 22 \), so \( B'D' = 22\times\frac{1}{2}=11 \). Wait, but let's recheck:

Wait, the dilation rule is \( D_{Q,\frac{1}{2}} \), so center \( Q \), scale factor \( \frac{1}{2} \). So for angle: \( \angle C'B'D' = \angle CBD = 51^\circ \) (angle preservation). For \( CQ \): Let \( CQ = y \), \( C'Q = y - 3 \). Since \( \frac{C'Q}{CQ}=\frac{1}{2} \), \( y - 3=\frac{1}{2}y \), \( \frac{y}{2}=3 \), \( y = 6 \), so \( CQ = 6 \). For \( B'D' \): \( BD = 22 \), so \( B'D' = 22\times\frac{1}{2}=11 \).

Answer:

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