Question

1. you can transform △wxy to △wxy by translating it and then performing a dilation centered at the origin. so, △wxy ~ △wxy. find the translation rule and the scale factor of the dilation.
2. simplify the scale factor and write it as a proper fraction, improper fraction, or whole number.
3. translation: (x, y) → ( , )
4. scale factor:

Explanation:

Step1: Find coordinates of corresponding points

From the graph, assume a point \(X(4, - 5)\) in \(\triangle WXY\) and its corresponding point \(X'(-4,9)\) in \(\triangle W'X'Y'\) after translation and dilation. First, find the translation rule.
Let the translation rule be \((x,y)\to(x + a,y + b)\).
For point \(W(5, - 6)\) (assuming coordinates from the graph, if \(W\) in the original triangle and \(W'(0,0)\) after translation (before dilation)).
\(x\) - coordinate: \(0=5 + a\Rightarrow a=-5\)
\(y\) - coordinate: \(0=-6 + b\Rightarrow b = 6\)
So the translation rule is \((x,y)\to(x - 5,y + 6)\)

Step2: Apply translation and then find scale factor

After translation, assume a point \(X(4,-5)\) becomes \(X_t(4 - 5,-5 + 6)=(-1,1)\) (translation). Let the scale factor be \(k\).
If the final image of \(X_t\) is \(X'(-4,9)\) (after dilation centered at the origin \((x,y)\to(kx,ky)\))
For \(x\) - coordinate: \(-4=k\times(-1)\Rightarrow k = 4\)
For \(y\) - coordinate: \(9=k\times1\Rightarrow k = 9\) (This is wrong. Let's use another approach. Let's take the length of a side.
Assume in \(\triangle WXY\), if \(W(5,-6)\), \(X(4,-5)\), \(Y(5,-9)\) (assuming from the graph). After translation \(W_t(0,0)\), \(X_t(-1,1)\), \(Y_t(0,-3)\)
In \(\triangle W'X'Y'\), if \(W'(0,0)\), \(X'(-4,9)\), \(Y'(0,-12)\)
The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
Length of \(W_tX_t\): \(d_{W_tX_t}=\sqrt{(-1 - 0)^2+(1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\)
Length of \(W'X'\): \(d_{W'X'}=\sqrt{(-4-0)^2+(9 - 0)^2}=\sqrt{16 + 81}=\sqrt{97}\) (Wrong approach. Let's use the ratio of coordinates from the origin - centered dilation.
Since dilation is centered at the origin. If a point \(P(x,y)\) is dilated to \(P'(kx,ky)\).
Take a non - origin point (after translation). Assume after translation a point \(X_t(x_1,y_1)\) and its image \(X'(x_2,y_2)\)
\(x_2=kx_1\), \(y_2=ky_1\)
If we consider the vertical side (assuming from the graph). Let's take the length from \(W\) (after translation \(W_t(0,0)\)) to \(Y\) (after translation \(Y_t\)). If in the original (after translation) \(Y_t\) has \(y\) - coordinate \(y_1=-3\) and in the dilated triangle \(Y'\) has \(y\) - coordinate \(y_2=-12\)
\(k=\frac{y_2}{y_1}=\frac{-12}{-3}=4\)

Answer:

Translation: \((x,y)\to(x - 5,y + 6)\), Scale factor: \(4\)