Question

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

Explanation:

Step1: Find coordinates of corresponding points

Let's take point \(B(- 8,1)\) in the original figure and \(B'(-8,8)\) in the dilated figure.

Step2: Calculate the scale factor

The formula for scale factor \(k\) in a dilation centered at the origin is \(k=\frac{y_{B'}}{y_{B}}\) (since \(x\) - coordinates are same for \(B\) and \(B'\) in this case, we can use \(y\) - coordinates. Also, for a dilation \((x,y)\to(kx,ky)\)).
\(k = \frac{8}{1}=8\)

Answer:

Translation: \((x,y)\to(x,y + 9)\) (because for point \(C(-6,-8)\) in original (purple) figure and \(C'(0,0)\) in the green figure, \(x\) changes from \(-6\) to \(0\) (a change of \(6\), but wait, no - re - check. Wait, original purple figure: \(B(-8, - 1)\), \(C(-6,-8)\), \(D(-4,-1)\); green figure: \(B'(-8,8)\), \(C'(0,0)\), \(D'(8,8)\). The translation: for \(x\) - coordinate: from \(x=-8\) (for \(B\)) to \(x = - 8\) (for \(B'\)) (no \(x\) - translation), for \(y\) - coordinate: \(y=-1\) (for \(B\)) to \(y = 8\) (for \(B'\)), so \(y\) - translation \(t_y=8-(-1)=9\). So translation \((x,y)\to(x,y + 9)\). Scale factor \(k = 8\) (using \(B(-8,-1)\) and \(B'(-8,8)\), \(\frac{8}{-1}=- 8\) (error in previous step, correct: original \(B(-8,-1)\), \(B'(-8,8)\). Since dilation formula is \((x,y)\to(kx,ky)\) after translation. Wait, no - the dilation is centered at the origin. Wait, first translation: move the purple figure up by \(9\) units (so \((x,y)\to(x,y + 9)\), then dilation. For the translated \(B(-8,-1+9)=(-8,8)\) (matches \(B'\)), translated \(C(-6,-8 + 9)=(-6,1)\), but after dilation \((-6,1)\to(-6k,1k)\) and \(C'(0,0)\) (error. Wait, no - original purple \(B(-8,-1)\), \(C(-6,-8)\), \(D(-4,-1)\); green \(B'(-8,8)\), \(C'(0,0)\), \(D'(8,8)\). First, translation: \((x,y)\to(x,y+9)\) (moves \(B(-8,-1)\to(-8,8)\), \(C(-6,-8)\to(-6,1)\), \(D(-4,-1)\to(-4,8)\). Then dilation: \((x,y)\to(kx,ky)\). For \(C(-6,1)\) to \(C'(0,0)\) (error. Wait, no - actually, the center of dilation is the origin. The ratio of distances from the origin. For \(B(-8,-1)\) (after translation \(x=-8,y = 8\) (wait no - confusion. Wait, correct approach:
Let’s use the formula for dilation \((x,y)\to(kx,ky)\). Take a point from the original (after translation) and the dilated figure.
Original (purple) \(B(-8,-1)\), after translation (to make it easier to see dilation from origin) assume translation \((x,y)\to(x,y + 9)\) gives \(B_t(-8,8)\), \(C_t(-6,1)\), \(D_t(-4,8)\). But \(C_t(-6,1)\) and \(C'(0,0)\) (no - wrong. Wait, actually, the dilation formula: if \(A(x,y)\) and \(A'(x',y')\) and dilation centered at origin \((x',y')=(kx,ky)\). Take \(B(-8,-1)\) and \(B'(-8,8)\), \(k=\frac{8}{-1}=-8\) (but lengths: distance from \(B\) to \(C\): \(\sqrt{(-6 + 8)^2+(-8 + 1)^2}=\sqrt{4 + 49}=\sqrt{53}\), distance from \(B'\) to \(C'\): \(\sqrt{(0 + 8)^2+(0 - 8)^2}=\sqrt{64+64}=\sqrt{128}\). Also, using \(D(-4,-1)\) and \(D'(8,8)\), \(k=\frac{8}{-1}=-8\). Translation: \((x,y)\to(x,y+9)\) (because \(y\) - coordinate of \(B\) is \(-1\) and \(B'\) is \(8\), \(8-(-1)=9\)). So translation \((x,y)\to(x,y + 9)\) and scale factor \(k=-8\)

Translation: \((x,y)\to(x,y + 9)\), Scale factor: \(-8\)