Dilation scale factor examples

Summary

The script explains the concept of dilation in geometry, focusing on how to determine the scale factor of a dilation and how it affects the lengths of corresponding sides between two figures. It demonstrates the process using two pentagons and a triangle, showing that the scale factor can be found by comparing the lengths of corresponding sides or by using the given scale factor to find unknown lengths. The examples illustrate the consistent application of the scale factor to transform one figure into another.

Takeaways

• 📐 The scale factor in a dilation is the ratio by which distances between corresponding points change.
• 🔍 To determine the scale factor, compare the lengths of corresponding sides or segments in the original and dilated figures.
• 📏 The scale factor can be found by dividing the length of a side in the dilated figure by the length of the corresponding side in the original figure.
• 🤔 If the center of dilation is not provided, the scale factor can still be determined by examining the distances between points.
• 📉 In the first example, the scale factor is 1/3, as the length of side AB in the original figure (6 units) is three times longer than in the dilated figure (2 units).
• 📈 The scale factor affects both the x and y changes in a figure's coordinates, as demonstrated by the changes from points A to E in the script.
• 🔢 The scale factor of 5/2 in the second example means that the lengths of the sides in the original figure are 5/2 times longer in the dilated figure.
• 📐 For the second example, the length of segment A'E' is calculated by multiplying the length of AE (2 units) by the scale factor (5/2), resulting in 5 units.
• 🔄 The scale factor applies to all corresponding sides and distances in a figure undergoing dilation.
• 📝 When the original figure is not drawn, the scale factor can be used to calculate the lengths of sides in the dilated figure.
• 🧩 In the third example, knowing the scale factor of two and the length of A'B' (8 units), the length of AB in the original figure can be found by dividing by the scale factor, resulting in 4 units.

Q & A

What is a dilation in geometry?

A dilation is a transformation that enlarges or reduces a figure by a certain scale factor, maintaining the shape of the figure but changing its size.

How can you determine the scale factor of a dilation without the center of dilation?

You can determine the scale factor by comparing the lengths of corresponding sides or points in the original and dilated figures. The scale factor is the ratio of the lengths of the corresponding sides or points.

What is the scale factor of the dilation for pentagon A'B'C'D'E' in the script?

The scale factor for the dilation of pentagon A'B'C'D'E' is 1/3, as determined by comparing the lengths of corresponding sides A to B and A' to B'.

How did the instructor calculate the scale factor using points A and B?

The instructor calculated the scale factor by observing that the distance between points A and B in the original pentagon is six units, and the corresponding distance in the dilated pentagon is two units, resulting in a scale factor of 1/3.

What is the relationship between the scale factor and the changes in coordinates of corresponding points?

The changes in the coordinates (both x and y) of corresponding points are scaled by the scale factor. For example, if the change in y for a point in the original figure is negative three, the change in y for the corresponding point in the dilated figure would be 1/3 of negative three.

How can you find the length of a segment in a dilated figure if you know the scale factor and the length of the corresponding segment in the original figure?

You multiply the length of the segment in the original figure by the scale factor to find the length of the corresponding segment in the dilated figure.

What is the length of segment A'E' in the second example where the scale factor is given as 5/2?

The length of segment A'E' is five units, as it is calculated by multiplying the original length of segment AE (which is two units) by the scale factor of 5/2.

In the third example, how did the instructor determine the length of segment AB without it being drawn?

The instructor determined the length of segment AB by knowing that the length of the dilated segment A'B' is eight units and that the scale factor is two. By dividing eight by two, the instructor found that the length of AB is four units.

What is the importance of the scale factor in determining the lengths of segments in dilated figures?

The scale factor is crucial as it dictates how much the lengths of segments in the original figure will be enlarged or reduced in the dilated figure. It is the multiplier used to calculate the new lengths.

Why is it not necessary to know the center of dilation to determine the scale factor or lengths of segments in dilated figures?

Knowing the center of dilation is not necessary because the scale factor is determined by the ratio of the lengths of corresponding segments or points, which can be observed directly from the figures without needing the center of dilation.