Question

triangle bcd was dilated using the rule ( d_{q, \frac{1}{2}} ). what are the values of the unknown measures? ( mangle cbd = ) (\boxed{}) ( ^circ ), ( cq = ) (\boxed{}), ( bd = ) (\boxed{}) (diagram shows triangle bcd with points c, b, d; dilated triangle cbd with center q, lengths: bc=36, bc=18, bd=22, dq=2, cq=3, angles: ∠c=34°, ∠d=51°)

Explanation:

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

Dilation is a similarity transformation, so corresponding angles of similar triangles are equal. In \( \triangle BCD \), we can find \( \angle CBD \) first. The sum of angles in a triangle is \( 180^\circ \). So \( \angle CBD = 180^\circ - 34^\circ - 51^\circ = 95^\circ \). Since \( \triangle C'B'D' \sim \triangle CBD \) (because of dilation), \( m\angle C'B'D' = m\angle CBD = 95^\circ \).

Step2: Find \( CQ \)

First, find the scale factor of dilation. The length of \( CB \) is 36 and \( C'B' \) is 18, so the scale factor \( k = \frac{18}{36} = \frac{1}{2} \), which matches the dilation rule \( D_{Q,\frac{1}{2}} \). Let \( CQ = x \) and \( C'Q = 3 \). The distance from \( C \) to \( Q \) and \( C' \) to \( Q \) should be related by the scale factor. Wait, actually, looking at the diagram, \( C'Q = 3 \), and since the scale factor is \( \frac{1}{2} \), let's see the relationship between \( CC' \) and \( C'Q \)? Wait, no, maybe better to use the fact that \( C' \) is the image of \( C \) under dilation with center \( Q \)? Wait, no, dilation center is \( Q \), so \( QC' = \frac{1}{2} QC \). Let \( QC = x \), then \( QC' = \frac{x}{2} \). But from the diagram, \( C'Q = 3 \), so \( \frac{x}{2} = 3 \), so \( x = 6 \)? Wait, no, maybe I misread. Wait, the length from \( C' \) to \( Q \) is 3, and since dilation factor is \( \frac{1}{2} \), then \( QC = 2 \times QC' = 2 \times 3 = 6 \)? Wait, but let's check the other side. \( CB = 36 \), \( C'B' = 18 \), which is \( \frac{1}{2} \) of 36, so scale factor is \( \frac{1}{2} \). So for the segment from \( C \) to \( Q \), the image is \( C' \) to \( Q \), so \( QC' = \frac{1}{2} QC \). So if \( QC' = 3 \), then \( QC = 6 \). Wait, but maybe the diagram has \( C'Q = 3 \), so \( CQ = 6 \)? Wait, let's check the other part. The length from \( D \) to \( D' \) is 2, so \( DD' = 2 \), and since scale factor is \( \frac{1}{2} \), \( QD' = \frac{1}{2} QD \). Let \( QD = y \), then \( QD' = \frac{y}{2} \), and \( DD' = QD - QD' = y - \frac{y}{2} = \frac{y}{2} = 2 \), so \( y = 4 \), so \( QD' = 2 \). But maybe that's not needed. Wait, back to \( CQ \). The length from \( C \) to \( C' \) plus \( C' \) to \( Q \) is \( CQ \). Wait, no, dilation center is \( Q \), so \( C \) is mapped to \( C' \), so \( Q \), \( C' \), \( C \) are colinear, with \( QC' = \frac{1}{2} QC \). So \( QC = QC' + CC' \)? No, \( C' \) is between \( Q \) and \( C \), so \( QC = QC' + C'C \)? Wait, no, if dilation center is \( Q \), then the image of \( C \) is \( C' \), so \( \overrightarrow{QC'} = \frac{1}{2} \overrightarrow{QC} \), meaning \( C' \) is halfway between \( Q \) and \( C \). So \( QC' = \frac{1}{2} QC \), so \( QC = 2 \times QC' = 2 \times 3 = 6 \). So \( CQ = 6 \).

Step3: Find \( B'D' \)

Since dilation scale factor is \( \frac{1}{2} \), and \( BD = 22 \), then \( B'D' = \frac{1}{2} \times BD = \frac{1}{2} \times 22 = 11 \).

Answer:

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