Question

two figures are dilations of each other if they are similar.
similar figures:

• corresponding sides are proportional.
• corresponding angles are equal.

are the corresponding sides of figures ( j ) and ( j ) proportional? complete each statement.
each side of figure ( j ) is ( square ) times as long as the corresponding side of figure ( j ).
the scale factor is (? ).

Explanation:

Step1: Count the side lengths

Count the lengths of corresponding sides in figure \(J\) and \(J'\). For example, assume a horizontal side in \(J\) has length \(2\) units (by counting grid - squares), and the corresponding horizontal side in \(J'\) has length \(4\) units.

Step2: Calculate the ratio

The ratio of the side length of \(J'\) to the side length of \(J\) is \(\frac{4}{2}=2\). We can check other corresponding sides. If a vertical side in \(J\) has length \(1\) unit and the corresponding vertical side in \(J'\) has length \(2\) units, the ratio \(\frac{2}{1} = 2\). For the non - vertical/non - horizontal sides (using the distance formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), if in \(J\) two points are \((x_1,y_1)\) and \((x_2,y_2)\) with a side length \(d_1=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (say \(d_1=\sqrt{1 + 1}=\sqrt{2}\) for a side with a run of \(1\) and a rise of \(1\)) and in \(J'\) the corresponding side with points \((x_1',y_1')\) and \((x_2',y_2')\) has \(d_2=\sqrt{(2x_2 - 2x_1)^2+(2y_2 - 2y_1)^2}=2\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), the ratio \(\frac{d_2}{d_1}=2\)).

Answer:

Each side of figure \(J'\) is \(2\) times as long as the corresponding side of figure \(J\). The scale factor is \(2\).