Question

in the similarity transformation of \\( \triangle abc \\) to \\( \triangle edf, \triangle abc \\) was dilated by a scale factor of ?, reflected across the , and moved through the translation .

Explanation:

Step1: Calculate the scale factor

First, find the length of a side in \(\triangle ABC\) and the corresponding side in \(\triangle EDF\).
Assume \(AB\): \(A(-4,-2)\), \(B(-2,-2)\), so \(AB=\vert-2 - (-4)\vert = 2\).
For \(DE\): \(D(3,1)\), \(E(5,1)\), so \(DE=\vert5 - 3\vert = 2\). Wait, no, better to use vertical or horizontal distances. Let's use \(BC\) and \(EF\). \(B(-2,-2)\), \(C(-2,-4)\), \(BC=\vert-2-(-4)\vert = 2\). \(E(5,1)\), \(F(4,0)\), \(EF\) is not good. Wait, use \(AB\) and \(DE\) wrong. Let's use \(AC\) and \(DF\). \(A(-4,-2)\), \(C(-2,-4)\), distance \(AC=\sqrt{(-2 + 4)^2+(-4 + 2)^2}=\sqrt{4 + 4}=\sqrt{8}\). \(D(3,1)\), \(F(4,0)\), distance \(DF=\sqrt{(4 - 3)^2+(0 - 1)^2}=\sqrt{1+1}=\sqrt{2}\). The scale factor \(k=\frac{DF}{AC}=\frac{\sqrt{2}}{\sqrt{8}}=\frac{1}{2}\). Another way: count grid units. \(AB\) is \(2\) units (from \(x=-4\) to \(x = - 2\) at \(y=-2\)). \(DE\) is \(2\) units (from \(x = 3\) to \(x=5\) at \(y = 1\)) no. Wait, wrong. Let's use \(BC\): \(B(-2,-2)\) to \(C(-2,-4)\) is \(2\) units. \(EF\): no. Wait, \(A(-4,-2)\), \(B(-2,-2)\), \(C(-2,-4)\). \(D(3,1)\), \(E(5,1)\), \(F(4,0)\). The length of \(AB\) (horizontal side): \(AB=\vert-2-(-4)\vert=2\). The length of \(DE\) (horizontal side): \(DE=\vert5 - 3\vert=2\) no. Wait, no, the vertical distance. \(B(-2,-2)\) to \(C(-2,-4)\) is \(2\) units. \(F(4,0)\) to \(D(3,1)\) (vertical - like) distance: \(\sqrt{(3 - 4)^2+(1 - 0)^2}=\sqrt{2}\). Wait, no, better: if we consider the transformation from \(\triangle ABC\) to \(\triangle EDF\). Let's assume \(A(-4,-2)\), \(B(-2,-2)\), \(C(-2,-4)\). \(D(3,1)\), \(E(5,1)\), \(F(4,0)\). If we dilate \(\triangle ABC\). Let's use the formula for dilation. Suppose the center of dilation is the origin (since no other info). The coordinates of \(A(-4,-2)\) after dilation \( (x,y)\to(kx,ky)\). If \(A(-4,-2)\) dilates to \(D(3,1)\) (after reflection and translation). Wait, no, first, ignore reflection and translation for scale - factor. Let's use \(B(-2,-2)\). If we assume dilation first. Suppose \(B(-2,-2)\) dilates to \(E(5,1)\) (after reflection and translation). No, better: count the number of units. The length of \(AB\) (horizontal) is \(2\) (from \(x=-4\) to \(x=-2\) at \(y = - 2\)). The length of \(DE\) (horizontal) is \(2\) (from \(x = 3\) to \(x=5\) at \(y=1\)) no. Wait, wrong approach. Let's use the ratio of side lengths. \(\triangle ABC\): \(AB = 2\) (horizontal), \(BC=2\) (vertical). \(\triangle EDF\): \(DE = 2\) (horizontal), \(DF=\sqrt{2}\) (from \(D(3,1)\) to \(F(4,0)\)). No, wait, the correct way: assume \(A(-4,-2)\), \(B(-2,-2)\), \(C(-2,-4)\). \(D(3,1)\), \(E(5,1)\), \(F(4,0)\). If we consider the transformation: first dilation. Let’s use the formula for dilation. Suppose we take a point \(A(-4,-2)\). After dilation by scale factor \(k\), reflection over \(x\) - axis (changes \(y\) sign) and translation \((x,y)\to(x + h,y + k)\). But another way: count the number of units from \(y\) - values. \(A(-4,-2)\) to \(D(3,1)\): vertical change from \(y=-2\) to \(y = 1\) (after reflection). If we assume dilation first. The distance from \(x\) - axis for \(A\) is \(2\), for \(D\) is \(1\). So scale factor \(k=\frac{1}{2}\).

Step2: Reflection

The \(y\) - coordinate of points in \(\triangle ABC\) is negative (\(A(-4,-2)\), \(B(-2,-2)\), \(C(-2,-4)\)) and in \(\triangle EDF\) is non - negative (\(D(3,1)\), \(E(5,1)\), \(F(4,0)\)). So it is reflected across the \(x\) - axis (the transformation \((x,y)\to(x,-y)\)).

Step3: Translation

After dilation (\((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\)) and reflection \((x,y)\to(x,-y)\): \(A(-4,-2)\): after dilation \((-2,-1)\), after reflection \((-2,1)\). To get to \(D(3,1)\), we use the translation \((x,y)\to(x + 5,y)\) (since \(-2+5=3\) and \(1=1\)).

Answer:

The scale factor is \(\frac{1}{2}\), reflected across the \(x\) - axis, and the translation is \((x,y)\to(x + 5,y)\)