Question

you can transform \\( \triangle t u v \\) to \\( \triangle t ^ { prime } u ^ { prime } v ^ { prime } \\) by translating it and then performing a dilation centered at the origin. so, \\( \triangle t u v \sim \triangle t ^ { prime } u ^ { prime } v ^ { 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 ) \mapsto ( square, square ) \\) scale factor: \\( \square \\)

Explanation:

Step1: Find the translation rule

• First, find the coordinates of a corresponding pair of points. Let's take point \(T(5,-4)\) and \(T'(-9,-6)\).
• For the \(x\) - coordinate: \(x\) - coordinate of \(T\) is \(5\), \(x\) - coordinate of \(T'\) is \(-9\). The change in \(x\) is \(-9 - 5=-14\).
• For the \(y\) - coordinate: \(y\) - coordinate of \(T\) is \(-4\), \(y\) - coordinate of \(T'\) is \(-6\). The change in \(y\) is \(-6-(-4)=-2\).
• The translation rule is \((x,y)\to(x - 14,y - 2)\).

Step2: Find the scale factor

• After translation, assume we have a point \(V\) with coordinates \((8,1)\). After translation \(V\) would be \((8-14,1 - 2)=(-6,-1)\). The image of \(V\) after dilation is \(V'(-4,9)\). But we can also use the ratio of the lengths of corresponding sides.
• Let's use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• For \(\triangle TUV\), take \(TU\): \(T(5,-4)\), \(U(9,-2)\). \(d_{TU}=\sqrt{(9 - 5)^2+(-2+4)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\).
• For \(\triangle T'U'V'\), take \(T'U'\): \(T'(-9,-6)\), \(U'(0,0)\). \(d_{T'U'}=\sqrt{(0 + 9)^2+(0 + 6)^2}=\sqrt{81+36}=\sqrt{117}=3\sqrt{13}\). Another way:
• Let's use the ratio of the \(y\) - coordinates (since dilation is centered at the origin after translation).
• If we consider the vertical distance from \(U\) to \(V\) in \(\triangle TUV\): \(y_V - y_U=1-(-2)=3\). In \(\triangle T'U'V'\), \(y_{V'}-y_{U'}=9 - 0 = 9\).
• The scale factor \(k=\frac{9}{3}=3\).

Answer:

Translation: \((x,y)\to(x - 14,y - 2)\)
Scale factor: \(3\)