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.

Outlines

00:00

📏 Understanding Scale Factors in Dilation

The paragraph explains the concept of dilation in geometry, specifically using a pentagon as an example. It describes how to determine the scale factor of a dilation when the center of dilation is not given. The instructor uses the distances between corresponding points of the original and dilated figures to illustrate how the scale factor affects these distances. The example provided shows a change from a length of six to two, indicating a scale factor of 1/3. The paragraph also explores how to handle cases where the line connecting points is neither vertical nor horizontal, using the changes in y and x coordinates to determine the scale factor. Additionally, it provides an example of calculating the length of a segment in a dilated figure when the scale factor is known.

Keywords

💡Dilation

Dilation in geometry refers to the transformation that enlarges or reduces a figure according to a certain ratio, known as the scale factor. In the video, the concept of dilation is central as it explains how a pentagon is transformed into another pentagon with a different size but the same shape. The script uses the example of pentagon ABCDE being dilated to form pentagon A'B'C'D'E'.

💡Scale Factor

The scale factor is the ratio by which dimensions of a figure are multiplied during a dilation. It is a key concept in the script, determining how much a figure is enlarged or reduced. The script explains that to find the scale factor, one must compare the lengths of corresponding sides of the original and dilated figures, as seen when comparing the lengths of sides AB and A'B'.

💡Corresponding Points

Corresponding points are points in the original and the dilated figures that occupy the same relative position in both figures. The script mentions corresponding points to illustrate how distances between these points change according to the scale factor, such as the distances between points A and B, and their corresponding points A' and B'.

💡Image of a Figure

The image of a figure is the result of a geometric transformation, such as a dilation, applied to the original figure. In the script, the pentagon A'B'C'D'E' is described as the image of pentagon ABCDE under a dilation, emphasizing the relationship between the original and the transformed figure.

💡Center of Dilation

The center of dilation is the fixed point around which a figure is scaled during a dilation. Although the script mentions that the center of dilation is not given, it is understood that all corresponding points in the original and dilated figures are connected by lines that pass through this center and are scaled by the same factor.

💡Change in Distance

Change in distance refers to the difference in the lengths between corresponding points before and after a dilation. The script uses the change in distance to illustrate how to determine the scale factor, such as the change from a length of six to a length of two between points A and B.

💡Vertical and Horizontal Lines

Vertical and horizontal lines are directions used to describe the orientation of lines or distances in a plane. The script discusses how changes in y (vertical) and x (horizontal) scale according to the dilation factor, which is essential for understanding how points are moved during dilation.

💡Length of Segment

The length of a segment is the measure of the distance between its endpoints. In the script, the length of segments such as AB and A'B' is used to demonstrate how to calculate the scale factor and the resulting lengths after dilation.

💡Negative Change

A negative change indicates a movement in the opposite direction on a coordinate plane, typically downward for changes in y. The script mentions a negative change in y to show how the vertical distance between points A and E is scaled by the dilation factor.

💡Multiplication by Scale Factor

Multiplication by the scale factor is the process of determining the new lengths of segments after dilation. The script explains that to find the length of a segment in the dilated figure, you multiply the original length by the scale factor, as demonstrated with the segment AE and its dilated counterpart A'E'.

💡Original Triangle

The original triangle is the figure before any transformation is applied. In the context of the script, the original triangle ABC is used to find the length of its sides before dilation, with the dilated triangle A'B'C' providing the scaled lengths for comparison.

Highlights

The concept of a dilation is introduced with a pentagon as an example.

Scale factor is essential to determine the change in distance between corresponding points during a dilation.

A clear example is given where the distance between points A and B changes from six to two, indicating a scale factor of 1/3.

The method to determine the scale factor is explained without the need for a center of dilation.

An alternative method to find the scale factor is presented using changes in y and x coordinates.

The scale factor of 1/3 is verified using the distance between points A and E.

A new example is introduced with a given scale factor of 5/2 to find the length of segment A'E'.

The length of A'E' is calculated by multiplying the original length of AE by the scale factor of 5/2.

The importance of the scale factor in determining the lengths of corresponding segments is emphasized.

A third example is presented involving a triangle with a scale factor of two to find the length of segment AB.

The length of AB is deduced by dividing the length of A'B' by the scale factor.

The process of finding lengths in dilated figures is simplified without needing to draw the entire image.

The transcript demonstrates the application of scale factors in various geometric dilation scenarios.

The importance of understanding the relationship between original and dilated figures is highlighted.

The transcript concludes by reinforcing the concept that corresponding sides are scaled by the scale factor.

A comprehensive explanation of dilation and scale factors is provided, suitable for educational purposes.

Transcripts

- [Instructor] We are told that pentagon

A'B'C'D'E',

which is in red right over here,

is the image of pentagon ABCDE

under a dilation.

So that's ABCDE.

What is the scale factor of the dilation?

So they don't even tell us the center of the dilation,

but in order to figure out the scale factor

you just have to realize when you do a dilation,

the distance between corresponding points

will change according to the scale factor.

So for example we could look at the distance

between point A and point B right over here.

What is our change in y?

Our change in, or even what is our distance?

Our change in y is our distance

because we don't have a change in x.

Well this is one, two, three,

four, five, six.

So this length right over here

is equal to six.

Now what about the corresponding side

from A' to B'?

Well this length right over here is equal to two,

and so you can see we went from having

a length of six to a length of two,

so you would have to multiply by 1/3.

So our scale factor right over here

is 1/3.

Now you might be saying okay that was pretty straightforward

because we had a very clear,

you could just see the distance between A and B.

How would you do it if you didn't have a vertical

or a horizontal line?

Well one way to think about it is,

the changes in y and the changes in x

would scale accordingly.

So if you looked at the distance

between point A and point E,

our change in y is negative three right over here,

and our change in x is positive three right over here.

And you can see over here between A' and E',

our change in y is negative one,

which is 1/3 of negative three,

and our change in x is one,

which is 1/3 of three.

So once again you see our scale factor

being 1/3.

Let's do another example.

So we are told that pentagon A'B'C'D'E'

is the image, and they don't,

they haven't drawn that here,

is the image of pentagon ABCDE

under a dilation with a scale factor of 5/2.

So they're giving us our scale factor.

What is the length of segment A'E'?

So as I was mentioning while I read it,

they didn't actually draw this one out.

So how do we figure out the length of a segment?

Well I encourage you to pause the video

and try to think about it.

Well they give us the scale factor,

and so what it tells us,

the scale factor is 5/2.

That means that the corresponding lengths will change

by a factor of 5/2.

So to figure out the length of segment A'E',

this is going to be,

you could think of it as the image

of segment AE.

And so you can see that

the length of AE

is equal to two.

And so the length of A'E'

is going to be equal to

AE which is two times the scale factor,

times 5/2, this is our scale factor right over here.

And of course what's two times 5/2?

Well it is going to be equal to five,

five of these units right over here.

So in this case we didn't even have to draw

A'B'C'D'E'.

In fact they haven't even given us enough information.

I could draw the scale of that,

but I actually don't know where to put it

because they didn't even give us

our center of dilation.

But we know that corresponding sides,

or the lengths between corresponding points,

are going to be scaled by the scale factor.

Now with that in mind,

let's do another example.

So we are told that triangle A'B'C',

which they depicted right over here,

is the image of triangle ABC,

which they did not depict,

under a dilation with a scale factor of two.

What is the length of segment AB?

Once again they haven't drawn AB here,

how do we figure it out?

Well it's gonna be a similar way as the last example,

but here they've given us the image

and they didn't give us the original.

So how do we do it?

Well the key, and pause the video again

and try to do it on your own.

Well the key realization here is

that if you take the length of segment AB

and you were to multiply by the scale factor,

so you multiply it by two,

then you're going to get the length

of segment A'B'.

The image's length is equal to the scale factor

times the corresponding length on our original triangle.

So what is the length of A'B'?

Well this is straightforward to figure out.

It is one, two, three, four,

five, six, seven, eight.

So this right over here is eight,

so we have two times the length of segment AB

is equal to eight.

And then you get the length of segment AB,

just divide both sides by two,

is equal to four.

And we're done.