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## 8th grade (Illustrative Mathematics)

### Unit 2: Lesson 5

Lesson 9: Side lengths quotients in similar triangles

# Side lengths after dilation

Sal dilates a shape and then compares the side lengths of the pre-image and image. Created by Sal Khan.

## Want to join the conversation?

• where did he get 5 from, and why is he squaring the numbers?
(4 votes)
• Hi 317354,
Sal is using something called the Pythagorean Theorem, used to find the missing side length of a triangle. (In this particular case, the actual triangle isn't a right angle but at , he draws an immaginary line from point P to point B to make a smaller right triangle within the original.)
In the Pythagorean Theorem, the missing side (you may have noticed he calls it the hypotenuse because that is the name of the line opposite the right angle) is found with the formula:
A² + B² = C²
So, following this formula, line AP which is 4 blocks long must be squared and added to line BP which is 3 blocks long squared to get to C²
This is 16 + 9 = C²
16 + 9 = 25 and the square root of 25 is 5 so line AB equals 5.
If you want to find out more about the Pythagorean Theorem, you're can visit https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/cc-8th-pythagorean-theorem/v/the-pythagorean-theorem on Khan Acadamy.
I hope this helps!
(5 votes)
• Instead of the Pythagorean theorem, how could you have used the distance formula here?

Because I don’t see how -4/3 would be helpful.
(3 votes)
• Instead of the Length of side AB…

-4/3 is the Slope of side AB
, so it's the Change in y divided by the Change in x, ∆y/∆x .

Which does include some of the info we need, but it's used differently…

⭐The Distance Formula is another way to use the Pythagorean Theorem.

It is applied similarly, just calculated by the Change in coordinates

Distance Formula:
√(∆x^2 + ∆y^2
)

take the Square Root of: (the Squared Change in x), plus, (the Squared Change in y).

Coordinates of Points…
A (3, 5),
B (6, 1)

√(∆x^2 + ∆y^2
)
=
√((x2 - x1)^2 + (y2 - y1)^2)
=
√((6 - 3)^2 + (1 - 5)^2)
=
√(3^2 + (-4)^2)
=
√(9 +16)
=
√25
=
5 ← 🥳 the hypotenuse of Right Triangle APB, the Distance between A and B, and the Length of line segment AB.

We know from the directions: Triangle of ABC is transformed by a scale factor of two, so…

its Image's AB side is: 2 • 5 = 10

⭐We can directly measure the Image's line segment AB by using its coordinates within the Distance Formula.

A' (3, 9)
B' (9, 1
)

√(∆x^2 + ∆y^2)
=
√((9 - 3)^2 + (1 - 9)^2)
=
√(6^2 + (-8)^2)
=
√(36 +64)
=
√100
=
10 ←🥳 the hypotenuse of the Image of right triangle APB, the Distance between A' and B', and the Length of the Image's line segment AB.

(≧▽≦) I hope this helps!
(5 votes)
• Are scale maps the only thing we would use dilations for or not?
(2 votes)
• Hi Samantha Fuller,

Here are some real world examples:

When you need to enlarge a picture from wallet-size to wall-hanging size, you could use dilation.

As you are in normal view of a scene on a camera, when you zoom in to see more of the picture you use dilation to do this.

Criminal investigators, especially on television, use dilation to see larger versions of evidence such as fingerprint scanning or photos.

When architects are trying to build a building with dimensions previously made out by a model, they have to increase the size of dimensions.

Our last example includes when our pupils in our eyes dilate because of the adjustments they are making to the amount of light intake.

Source: A Prezi by Victoria Gray

Hope this helps!
- Sam
(4 votes)
• At , why did we go 4 points above?
(3 votes)
• Because the point is 4 above P, so we move the point up 4.
(1 vote)
• so we could multiply are first answer by 2?
(3 votes)
• Would you have to understand 'squaring' numbers to firgure out this lesson?
(2 votes)
• Not necessary but it would be much easier if you understood.
(2 votes)
• Side lengths after dilation
(2 votes)
• I am trying to go in order of units but after this video there is nothing what do I do??
(2 votes)
• Please explain how the following sequences of transformations to a shape will definitely result in a congruent shape?
(X,Y)-----(7X, 7Y)
(X,Y)-----(X+3, Y-1)
(2 votes)
• Before the video ended do you leave the triangle big or leave it small
(2 votes)

## Video transcript

The graph below contains triangle ABC and the point P. Draw the image of triangle ABC under a dilation whose center is P and a scale factor of 2. So essentially, we want to scale this so that every point is going to be twice as far away from P. So for example, B right over here has the same y-coordinate as P, but its x-coordinate is three more. So we want to be twice as far. So if this maps to point B, we just want to go twice as far. So we're at 3 away, we want to go 6 away. So point P's x-coordinate is at 3, now we're at 9. Likewise, point C is 3 below P. Well we want to go twice as far, so we'll go 3 more. And point A is 4 above P. Well we want to go 4 more. We want to go twice as far-- one, two, three, four. And we get right over there. Then they ask us, what are the lengths of side AB and its image? AB right over here, let's see, we might have to apply the distance formula. Let's see, it's the base right over here. The change in x between the two is 3 and the change in y is 4, so this is actually a 3, 4, 5 right triangle. 3 squared plus 4 squared is equal to 5 squared. So AB is 5 units long. Essentially just using the Pythagorean theorem to figure that out. And its image, well it's image should be twice as long. And let's see whether that actually is the case. So this is a base right over here that's of length 6. This has a height, or this change in y, I could say. Because I'm really just trying to figure out this length, which is the hypotenuse of this right triangle. I don't have my drawing tool, so I apologize. But this height right here is 8. So 8 squared is 64, plus 6 squared is 36, that's 100, which is 10 squared. So notice, our scale factor of 2, the corresponding side got twice as long. Each of these points got twice as far away from our center of dilation.