Question

you can transform \\( \triangle b c d \\) to \\( \triangle b ^ { prime } c ^ { prime } d ^ { prime } \\) by translating it and then performing a dilation centered at the origin. so, \\( \triangle b c d \sim \triangle b ^ { prime } c ^ { prime } d ^ { prime } \\). find the translation rule and the scale factor of the dilation. simplify the scale factor and write it as a proper fraction, improper fraction, or whole number. translation: \\( ( x, y ) \
ightarrow ( \square, \square ) \\) scale factor: \\( \square \\)

Explanation:

Step1: Find the translation rule

First, find the coordinates of a point before and after translation. Let's take point \(C(1,10)\) and assume after translation (before dilation) it maps to \(C'(0,0)\) (since dilation is centered at the origin).
The translation rule for \(x\) - coordinate: \(x\to x - 1\) (because \(1-1 = 0\)).
The translation rule for \(y\) - coordinate: \(y\to y - 10\) (because \(10 - 10=0\)). So the translation rule is \((x,y)\to(x - 1,y - 10)\)

Step2: Find the scale factor

Let's use another pair of corresponding points. Take \(B(6,5)\). After translation \(B\) becomes \((6 - 1,5 - 10)=(5,- 5)\). After dilation, \(B'\) is \((9,-9)\)
Let the scale factor be \(k\). If a point \((x,y)\) is dilated centered at the origin to \((kx,ky)\).
We have \(k\times5 = 9\) and \(k\times(-5)=-9\). Solving \(k=\frac{9}{5}\) (using the \(x\) - coordinate: \(\frac{9}{5}\times5 = 9\))

Answer:

Translation: \((x,y)\to(x - 1,y - 10)\)
Scale factor: \(\frac{9}{5}\)