Question

you can transform △bcd to △bcd by translating it and then performing a dilation centered at the origin. so, △bcd ~ △bcd. 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) → (, )

Explanation:

Step1: Find the translation rule

To find the translation rule, we can compare the coordinates of a corresponding point before and after translation (before dilation). Let's take point \(C(0,10)\) and \(C'(0,0)\).
The \(x\) - coordinate of \(C\) is \(0\) and of \(C'\) is \(0\), the \(y\) - coordinate of \(C\) is \(10\) and of \(C'\) is \(0\). So the translation rule is \((x,y)\to(x,y - 10)\).

Step2: Find the scale factor

After translation, we can use the ratio of the lengths of corresponding sides or the ratio of the coordinates of corresponding points (after translation) for dilation. Let's use point \(B(6,5)\) (after translation \(B\) would be \((6,5 - 10)=(6,- 5)\)) and \(B'(6,-10)\).
The scale factor \(k\) for dilation centered at the origin is given by the formula \(k=\frac{y_{B'}}{y_{B - translated}}\) (or \(k = \frac{x_{B'}}{x_{B - translated}}\) since \(x_{B - translated}=x_{B'}=6\)).
\(k=\frac{-10}{-5}=2\)

Answer:

Translation: \((x,y)\to(x,y - 10)\)
Scale factor: \(2\)