Question

problem 4
precisely describe a dilation that takes the smaller figure to the larger figure.

Explanation:

Step 1: Find the center of dilation

The center of dilation is the common point (the point that doesn't move during dilation). Looking at the figure, the center of dilation is the point \(C\).

Step 2: Calculate the scale factor

Pick a pair of corresponding points. For example, take a vertex of the smaller diamond \((1,0)\) and its corresponding vertex on the larger diamond \((- 2,2)\).
The distance from the center \(C=(2, - 1)\) to the point \((1,0)\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) is \(d_1=\sqrt{(2 - 1)^2+(-1 - 0)^2}=\sqrt{1 + 1}=\sqrt{2}\).
The distance from the center \(C=(2,-1)\) to the point \((-2,2)\) is \(d_2=\sqrt{(2+2)^2+(-1 - 2)^2}=\sqrt{16 + 9}=5\).
Another way (since the figures are on a grid and we can count the units of displacement from the center).
Let's use the vertical or horizontal distances.
Take a horizontal - like displacement:
For a point on the smaller figure: say from the center \(C=(2,-1)\) to \((1, - 1)\) (moving horizontally) is \(1\) unit.
For the corresponding point on the larger figure: from the center \(C=(2,-1)\) to \((-2,-1)\) (moving horizontally) is \(4\) units.
Take a vertical - like displacement: from the center \(C=(2,-1)\) to \((2,0)\) (moving vertically) is \(1\) unit. From the center \(C=(2,-1)\) to \((2,2)\) (moving vertically) is \(3\) units.
We can also use the ratio of side lengths.
Count the number of units for a side of the smaller diamond. The side length of the smaller diamond (using the distance between \((1,0)\) and \((2,-1)\)): \(s_1=\sqrt{(2 - 1)^2+(-1 - 0)^2}=\sqrt{2}\).
The side length of the larger diamond (using the distance between \((-2,2)\) and \((2, - 6)\)): \(s_2=\sqrt{(2 + 2)^2+(-6 - 2)^2}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\). But a simpler way is to note that if we consider the fact that if we assume the transformation is a dilation.
Let’s use the formula for dilation \((x,y)\to(k(x - h)+h,k(y - v)+v)\) where \((h,v)\) is the center of dilation.
Take a point \((x_1,y_1)\) on the smaller figure and \((x_2,y_2)\) on the larger figure. Let’s take \((1,0)\) (smaller) and \((-2,2)\) (larger) and center \((h = 2,v=-1)\)
\(x_2=k(x_1 - h)+h\) and \(y_2=k(y_1 - v)+v\)
For \(x\) - coordinate: \(-2=k(1 - 2)+2\), \(-2=-k + 2\), \(k = 4\)
For \(y\) - coordinate: \(2=k(0+1)-1\), \(2=k - 1\), \(k = 3\). Wait, this is wrong. Let's use another pair of points.
Take the top - most point of the smaller figure \((2,0)\) and the top - most point of the larger figure \((2,6)\)
Using \(y_2=k(y_1 - v)+v\), where \(y_1 = 0,y_2 = 6,v=-1\)
\(6=k(0 + 1)-1\), \(k=3\)
Take the left - most point of the smaller figure \((1,-1)\) and the left - most point of the larger figure \((-2,-1)\)
Using \(x_2=k(x_1 - h)+h\), where \(x_1 = 1,x_2=-2,h = 2\)
\(-2=k(1 - 2)+2\), \(-2=-k + 2\), \(k = 4\). Wait, no. Let's count the number of units from the center.
From the center \((2,-1)\) to a vertex of the smaller diamond (e.g., \((2,0)\)) is \(1\) unit up. From the center \((2,-1)\) to a vertex of the larger diamond (e.g., \((2,6)\)) is \(3\) units up.

Answer:

A dilation with center \((2,-1)\) and scale factor \(3\) takes the smaller figure to the larger figure.