Question

figures x, y, and z are similar.
which statement is true?
figure x is a dilation of figure y by a scale factor of 2.
figure y is a dilation of figure z by a scale factor of \\( \frac { 1 } { 2 } \\).
figure x is a dilation of figure z by a scale factor of 3.
figure z is a dilation of figure x by a scale factor of \\( \frac { 1 } { 4 } \\).

Explanation:

Step1: Understand the concept of dilation

Dilation is a transformation that changes the size of a figure. If a figure \(A\) is a dilation of figure \(B\) by a scale factor \(k\), then the ratio of the corresponding side lengths of \(A\) and \(B\) is \(k\). Let's assume we can count the side - lengths (using the grid). Suppose the side - length of \(Z\) is \(x\), the side - length of \(Y\) is \(2x\), and the side - length of \(X\) is \(4x\).

Step2: Check the scale factor for each option

• Option 1: Figure \(Z\) is a dilation of Figure \(X\) by a scale factor of \(\frac{1}{4}\)

If \(X\) is the original and \(Z\) is the dilated figure, the scale factor \(k=\frac{\text{side length of }Z}{\text{side length of }X}\). If side length of \(X = 4\) units and side length of \(Z = 1\) unit (by counting on the grid, assume a simple case), \(k=\frac{1}{4}\). But dilation is usually from the smaller to the larger or vice - versa. The formula for dilation is \(A = k\times B\). If \(Z=k\times X\), \(k=\frac{Z}{X}\). But if we consider the order of dilation (the center of dilation is fixed), when we go from \(X\) to \(Z\), it's a reduction.

• Option 2: Figure \(X\) is a dilation of Figure \(Z\) by a scale factor of \(3\)

If side length of \(Z = 1\) and side length of \(X = 4\), \(k=\frac{X}{Z}=4
eq3\)

• Option 3: Figure \(Y\) is a dilation of Figure \(Z\) by a scale factor of \(\frac{1}{2}\)

If side length of \(Z = 1\) and side length of \(Y = 2\), \(k = 2
eq\frac{1}{2}\)

• Option 4: Figure \(X\) is a dilation of Figure \(Y\) by a scale factor of \(2\)

If side length of \(Y = 2\) and side length of \(X = 4\), using the formula \(X=k\times Y\), then \(k=\frac{X}{Y}\). Substituting the values (by counting the number of grid units for corresponding sides), if \(Y\) has a side - length of \(2\) units (on the grid) and \(X\) has a side - length of \(4\) units (on the grid), \(k = 2\)

Answer:

Figure \(X\) is a dilation of Figure \(Y\) by a scale factor of \(2\).