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Lesson 7: Congruence and similarity

# Side lengths after dilation

In a dilation, if the scale factor is greater than 1, the shape will stretch away from the center of dilation, getting larger. If the scale factor is between 0 and 1, the shape will shrink towards the center, getting smaller. All the lengths in the figure change by the same scale factor. Created by Sal Khan.

## Want to join the conversation?

• 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.
• 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!
• Are scale maps the only thing we would use dilations for or not?
• 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
• where did he get 5 from, and why is he squaring the numbers?
• 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.
I hope this helps!
• At , why did we go 4 points above?
• Because the point is 4 above P, so we move the point up 4.
• Would you have to understand 'squaring' numbers to firgure out this lesson?
• Not necessary but it would be much easier if you understood.
• so we could multiply are first answer by 2?
• How do you know when to do the Pythagorean theorem on the triangle?
• If you know the lengths of two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the third side.

Have a blessed, wonderful day!