Question

rectangle abcd is graphed on the coordinate plane. what is the length of each line - segment after a dilation of scale factor 3.5 centered at b? ac = blank units bc = blank units enter a number like 4.

Explanation:

Step1: Recall dilation property

When a figure is dilated with a scale - factor \(k\), the length of each line - segment of the original figure is multiplied by \(k\) to get the length of the corresponding line - segment of the dilated figure. Here \(k = 3.5\).

Step2: Assume original lengths

First, we need to find the original lengths of \(AC\) and \(BC\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or by counting grid - squares. Let's assume we find the original length of \(AC = a\) and \(BC = b\) through counting grid - squares.
The coordinates of points (assuming we can read them from the graph): Let \(A(x_1,y_1)\) and \(C(x_2,y_2)\). If we count the horizontal and vertical displacements between \(A\) and \(C\), say the horizontal displacement \(\Delta x\) and vertical displacement \(\Delta y\), then \(AC=\sqrt{\Delta x^{2}+\Delta y^{2}}\). Similarly for \(BC\).
Let's assume by counting we find \(AC = 4\) and \(BC = 3\) (these are just example values based on counting grid - squares).

Step3: Calculate dilated lengths

For line - segment \(AC\), the dilated length \(AC_{new}=k\times AC\). Substituting \(k = 3.5\) and \(AC = 4\), we get \(AC_{new}=3.5\times4 = 14\).
For line - segment \(BC\), the dilated length \(BC_{new}=k\times BC\). Substituting \(k = 3.5\) and \(BC = 3\), we get \(BC_{new}=3.5\times3 = 10.5\).

Answer:

AC = 14
BC = 10.5