Pentagon A’B’C’D’E’ is the image of pentagon ABCDE after a dilation centered at F. What is the scale factor of this dilation? The scale factor is [blank1]
The Correct Answer and Explanation is:
The scale factor is 53\frac{5}{3}.
To determine the scale factor of a dilation, we compare the distances from the center of dilation to corresponding points on the original and image figures. In this case, point F is the center of dilation. The lengths from F to the original pentagon ABCDE are as follows:
- FA = 6
- FB = 3
- FC = 6
- FD = 3
- FE = 6
The corresponding distances from F to the dilated pentagon A′B′C′D′E′ are:
- FA′ = 10
- FB′ = 5
- FC′ = 10
- FD′ = 5
- FE′ = 10
To find the scale factor, we divide any distance from F to a vertex on the dilated figure by the corresponding distance from F to the same vertex on the original figure. For example:
Scale factor=FA′FA=106=53\text{Scale factor} = \frac{FA’}{FA} = \frac{10}{6} = \frac{5}{3}
Each pair of corresponding distances gives the same ratio, confirming that the transformation is a true dilation.
This scale factor of 53\frac{5}{3} means that every point on the original pentagon is moved away from point F by a factor of five-thirds. The new figure is similar to the original in shape but larger in size. Dilation preserves the shape and angle measures of geometric figures while altering their size proportionally.
Understanding dilation is essential in various applications such as computer modeling, architectural scaling, and map making, where proportionate resizing is required without distortion of form. The key principle is that distances are multiplied by the same factor from a fixed point, ensuring uniform scaling across all dimensions. This preserves similarity and allows for predictable transformations in two-dimensional and three-dimensional spaces.
