Question

rectangle abcd is graphed on the coordinate plane. its image after a dilation is efgh. (a) what is the scale factor of the dilation? enter a fraction like 5 in the box.

Explanation:

Step1: Select corresponding side lengths

Let's consider the side - length of the original rectangle \(ABCD\) and the dilated rectangle \(EFGH\). For example, if we look at the horizontal side of rectangle \(ABCD\) along the \(x\) - axis, assume the length of a side of rectangle \(ABCD\) is from \(x = 0\) to \(x = 1\), so its length \(l_1=1\). For the corresponding side of rectangle \(EFGH\) along the \(x\) - axis, it is from \(x=- 4\) to \(x = 4\), so its length \(l_2 = 8\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the dilated figure to the length of the corresponding side of the original figure. \(k=\frac{l_2}{l_1}\). Substituting \(l_1 = 1\) and \(l_2 = 8\) into the formula, we get \(k = 2\). We can also check with vertical sides. The vertical side of rectangle \(ABCD\) from \(y = 0\) to \(y = 1\) (length \(1\)) and the vertical side of rectangle \(EFGH\) from \(y=-4\) to \(y = 4\) (length \(8\)). The scale factor \(k=\frac{\text{length of side of }EFGH}{\text{length of side of }ABCD}\). If we take the distance between two corresponding vertices. For example, if we consider the distance from the origin to vertex \(A(1,0)\) in rectangle \(ABCD\) and the distance from the origin to vertex \(E(4,0)\) in rectangle \(EFGH\). The distance \(d_1\) from \((0,0)\) to \((1,0)\) is \(1\) and the distance \(d_2\) from \((0,0)\) to \((4,0)\) is \(4\). The scale factor \(k = 2\).

Answer:

2