Scale Factors
Objective
SO, HOW DO WE DILATE?
Well, the easiest way to do it by hand (when computer software is not available), is by drawing lines and increasing (or decreasing) the distance from our point to the new one.
This is one of those times when explaining the process is much harder than showing the process.
EXAMPLE 1:
So, first, in order to dilate a shape, we need a shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to duplicate those segments from the point on the triangle, away from our original point.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is doubled.
This is because we expanded our dilation two times.
So this triangle is twice as big as our first triangle.�
However, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
Just keep that in mind for later on…..�(Epic geometry foreshadowing)
EXAMPLE 2:
So, let’s do another example for whole number dilation.
Again, First, we’re going to need our shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to duplicate those segments from the point on the triangle, away from our original point.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is tripled.
This is because we expanded our dilation three times.
So this triangle is thrice as big as our first triangle.�
Again, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
Just an important note.
So, that’s whole number dilation, what about fractional dilation?
FRACTIONAL DILATION
So when we dilate a shape by a whole number it makes it bigger.
So what do you think would happen if we broke a shape into a fractional portion?
It shrinks!
And of course, depending on how small the fraction is, that is how much smaller the shape will be.
For example:
Example 3:
So, let’s do an example for fractional dilation.
Again, First, we’re going to need our shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to find the points on those segments that are half way away from our triangle.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is halved.
This is because we shrank our dilation a half times.
So this triangle is half as big as our first triangle.�
Again, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
So, now we can get to our activity
Distance Formula
So, let’s say we have a segment (that we’ve maybe added together?), and we would like to find out how long the segment is.
Well, we know that when we are trying to find the distance along a line, it’s usually in a unit of distance (like inches, centimeters, etc.). We also know, we can use a ruler to find the distance between two points on a line, like points G and H.
So, let’s say we take a ruler and try to find the distance between points G and H
Well, we know that the distance that we want to find is actually the absolute value of the difference of the two numbers on the ruler, which corresponds to the two endpoints.
We say absolute value because there’s no such thing in normal day to day conversation as a negative measurement.
So, then we know that:
So, that’s great if it’s an easy thing to measure right?
But what happens when it’s not? Well, we actually have a way to find that out, but first, let’s look at the Pythagorean Theorem (I promise this is not random).
So, the Pythagorean Theorem states: a2 + b2 = c2 when a and b are the sides of a triangle, and c is the hypotenuse.
Well, let’s say we have a line segment:
K B
And let’s say the coordinates for K are (0,1)
and the coordinates for B are (2,8)
Now, we want to know the distance of KB, but how can we do it?
What if we thought of KB as the hypotenuse of a triangle?
Well, this would be a lot easier to see if we had a graph right?
SO LET’S GRAPH IT!
Remember, we said that the point K was (0,1)
And the point B was (2,8)
Now, let’s see if we can’t make this into a triangle so we can use the Pythagorean theorem to figure out the length of KB
So, first, since we need KB to be the hypotenuse in order to work, we need to create a few sides.
Let’s go ahead and extend the bottom side from K straight to under B.
We’ll label this point E.
Then, we go ahead and connect E to B.
There! Now we have a triangle!
Well, now, since both the legs of the triangle are flat, we can see how long each leg is.
So, we can see that KE travels from 0 to 2, so we know that |2-0| = 2
And we can also see that EB travels from 1 to 8, so we know that |1 - 8| = |-7| = 7
So, now we can figure out the length of our hypotenuse!
So, we take: 22 + 72 = c2
Or:
4 + 49 = c2
53 = c2
Now we take the square root and we find that:
WOW, THAT WAS A LOT OF WORK
We found the distance, but that was a lot of work. Maybe there is an easier way?
Well, turns out there is!
Let’s think about what we just did though to make sense of it.
We figured out that KE was 2 units long because we took the x-coordinate in K and subtracted it from the x-coordinate in E.
We also figured out that EB was 7 units long because we took the y-coordinate of E and subtracted it from B.
Well, let’s see if we can’t put that into our Pythagorean Theorem, and find out what it spits out.
DERIVING THE DISTANCE FORMULA
So, essentially what we did was we took: x2 - x1 and we made that the length of a
Then, we took: y2 - y1 and we made that the length of b.
Now remember, the Pythagorean Theorem is: a2 + b2 = c2
Replacing a with x2 - x1 and b with y2 - y1 we get:
(x2 - x1)2 + (y2 - y1)2 = c2
Now, since we want the actual length of c, let’s take the square root, and what we are left with is:
SO THAT WAS A LOT OF REVIEW
But now we can talk about scale factors.
So, what’s a scale factor?
Well, a scale factor is:
“a number which multiplies (“scales”) a quantity”
So, when we deal with a scale multiple, it means we’re multiplying a part of the shape by a number.
This is different from a dilation however, because a dilation is multiplying every part of the shape by a number.
So what happens when we deal with scale factors?
Well…..
Applying the Scale Factor
Again, this is one of those things that is easier to show than to explain.
So, imagine we have a graph:
And in that graph, we have a shape
(Let’s say a square):
As we can see, that shape’s points are:
A(0,3) B(3,3) C(3,0) D(0,0)
But, what would happen if we multiplied all of the x coordinates by a factor of 3?
Well, then our new coordinates would be:
A(0,3) B(9,3) C(9,0) D(0,0)
Like so:
So, it stretched the shape by a factor of 3.
But, what if instead we multiplied all of the y coordinates by a factor of 3?
Well, then our new coordinates would be:
A(0,9) B(3,9) C(3,0) D(0,0)
So, this is what multiplying by a scale factor means.
But, when we do this
What does this do to the perimeter of the object?
Finding the Perimeter of a Dilated shape
To start off, let’s look at a shape that has been dilated by a certain factor first.
Then we can look at shapes that have only parts of them dilated by a scale factor.
So, first, we need an original shape:
And now, let’s measure the sides:
Now we know that the perimeter of this shape is going to be:
3 + 3 + 3 + 3
= 12
But what if we dilated this shape by a scale factor.
Say, 3?
Well, then we would have:
And now, let’s measure the sides:
Now we know that the perimeter of this shape is going to be:
9 + 9 + 9 + 9
= 36
And taking the area of the bigger shape, divided by the smaller shape gives us:
So, the perimeter is going to be 3 times the original perimeter.
So now we know we can multiply the original perimeter by the scale factor to get the new perimeter.
But what if it’s not the whole thing?
Well, to start, we first need an object:
So, as we can see, the lengths of each side of our square is 3.
And since the perimeter is just the sum of the sides of the object
Then we know the perimeter will be:
3 + 3 + 3 + 3
= 12
But what about when we multiplied all of the x-coordinates by a factor of 3?
What happens to the perimeter?
Well, let’s look:
Again, we know the perimeter will be:
9 + 3 + 9 + 3
= 24
And that seems a little weird, but let’s look at what we did.
We added an extra 6 units to each side
But left 2 of the sides the same.
So, if we were to write that out, it would look something like:
12 + 2(9 – 3)
So, the formula that we are looking for, would be:
= 12 + 2(6)
= 12 + 12
= 24
And that’s how we find the new perimeter
So, the way we can find the perimeter for a shape that has been dilated by a certain scale factor is:
Scale factor * original perimeter
�However, if only part of the shape has been dilated, then we can find the new perimeter by:
(Perimeter of the first shape) + 2(new length - old length)
But now, what does this mean for area?
FINDING THE AREA OF A DILATED SHAPE
To start off, let’s look at a shape that has been dilated by a certain factor first.
Then we can look at shapes that have only parts of them dilated by a scale factor.
So, first, we need an original shape:
And now, let’s measure the sides:
Now we know that the area of this shape is going to be:
3 * 3
= 9
But what if we dilated this shape by a scale factor.
Say, 3?
Well, then we would have:
And now, let’s measure the sides:
Now we know that the area of this shape is going to be:
9 * 9
= 81
And taking the area of the bigger shape, divided by the smaller shape gives us:
But that doesn’t seem to work because our scale factor is 3.
But what is 9?
Isn’t it 3 squared?
So, to find the new area, we take the scale factor, square it, then multiply it by the original area!
BUT WHAT IF IT’S NOT THE WHOLE THING THAT IS DILATED?
Well, to start, we first need an object:
So, as we can see, the lengths of each side of our square is 3.
And since the area is just the base times the height.
Then we know the area will be:
3 * 3
= 9
But what about when we multiplied all of the x-coordinates by a factor of 3?
What happens to the area?
Well, let’s look:
Again, we know the area will be:
9 * 3
= 27
And now, if we take our new area, and divide it by the original area, we get:
So, what this tells us is:
If the shape has only a part of it dilated by a scale factor
To find the new area
We multiply the old area by the scale factor.
Or:
AND THAT’S HOW WE FIND THE NEW AREA
So, the way we can find the area for a shape that has been dilated by a certain scale factor is:
However, if only part of the shape has been dilated, then we can find the new area by:
(Area of the first shape) * Scale Factor
So now, let’s look at Volume!
Finding the Volume of a Dilated 3-d shape
For volume, we’re only going to look at shapes that are dilated throughout the entire shape.
Mainly because if we talk about only dilating a part of the shape, it can get really complicated.
So to start, we need a shape:
And now, let’s measure the sides:
Now we know that the volume of this shape is going to be:
5 * 2 * 3
= 30
But what if we dilated this shape by a scale factor.
Say, 4?
Well, then we would have:
And now, let’s measure the sides:
Now we know that the volume of this shape is going to be:
20 * 8 * 12
= 1920
And taking the volume of the bigger shape, divided by the smaller shape gives us:
But that doesn’t seem to work because our scale factor is 4.
But what is 64?
Isn’t it 4 cubed?
So, to find the new volume, we take the scale factor, cube it, then multiply it by the original volume
So now, let’s do some examples:
5
3
2
20
12
8
Example 1:
Find the new perimeter of the shape if the x-coordinates of the shape are multiplied by 2.
So, we have a few ways of finding this out.
We can multiply each x-coordinate by 2
Then use the distance formula to find the new distance of each side
Then add all of the sides together
Or
We can use the formula:
So:
Example 2:
Find the new area of the shape if the y-coordinates of the shape are multiplied by 6.
So, we have a few ways of finding this out.
We can multiply each x-coordinate by 6
Then use the distance formula to find the new distance of each side
Then find the area
Or
We can use the formula:
So:
Example 3:
Find the new Volume of the shape if the shape is multiplied by 7.
So, for this one, we need to use the formula:
So first, let’s find the volume of this shape:
12
7.2
9.6
3
Now, we plug it in, and get: