Question

a paper clip is shown on the graph. the distance from a to b represents the length of the paper clip. dilate the paper clip by a scale factor of 2 with the center of dilation at the origin. complete the statement. point a is 7 cm up from point b, so point a is 7 cm up from point b. distance a to b = distance a to b · scale factor

Explanation:

Step1: Understand the property of dilation

Dilation is a transformation that changes the size of a figure. When we dilate a figure with a scale factor \(k\) and the center of dilation at the origin, the distance between any two points in the original figure and the corresponding points in the dilated figure is related by the formula \(d(A',B')=k\times d(A,B)\). In this case, the vertical distance between \(A\) and \(B\) is a part of the figure - specific distance. Since the scale factor \(k = 2\), but wait, no! Wait, looking at the graph (counting the vertical units from \(B\) to \(A\)):
If we count the vertical units from \(B\) (on the \(x -\)axis, \(y = 0\)) to \(A\) (where \(y=3\), but wait no - looking at the grid: each square is \(1\) cm. From \(B\) (at \(y = 0\)) to \(A\) (at \(y = 3\)) is \(3\) cm. But wait, no - wait the first box is filled with \(7\) which is wrong. Wait, no - re - check:
Wait, actually, when we dilate a figure with a scale factor \(k\) centered at the origin, if we have two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\), the corresponding points \(A'(kx_1,ky_1)\) and \(B'(kx_2,ky_2)\). The vertical distance \(d(A,B)=y_1 - y_2\) (assuming \(y_1>y_2\)) and \(d(A',B')=ky_1-ky_2=k(y_1 - y_2)\).
Counting the vertical units from \(B\) (lower point) to \(A\) (upper point) on the coordinate - plane (each grid square is \(1\) cm). The vertical distance from \(A\) to \(B\) is \(3\) cm. But the formula \(d(A',B')=d(A,B)\times\) scale factor. Given scale factor \(k = 2\).

Step2: Calculate the distance

If \(d(A,B)\) (vertical distance) is \(3\) cm (by counting the number of unit - squares between \(A\) and \(B\) vertically), and the scale factor \(k = 2\). Then \(d(A',B')=3\times2=6\) cm.

Answer:

The first box should be \(3\) and the second box (using the formula \(d(A',B')=d(A,B)\times\) scale factor, with scale factor \(2\)) should be \(6\)