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8th grade (Illustrative Mathematics)
Course: 8th grade (Illustrative Mathematics) > Unit 2
Lesson 5: Lesson 9: Side lengths quotients in similar trianglesSide 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.
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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.(5 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!(11 votes)
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 at1:06, 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!(4 votes)
Are scale maps the only thing we would use dilations for or not?(3 votes)![duskpin ultimate style avatar for user [SKLZ] ▁ ▂ ▄ ▅ ▆ ▇ █ נк █ ▇ ▆ ▅ ▄ ▂ ▁](https://cdn.kastatic.org/images/avatars/svg/duskpin-ultimate.svg)
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(5 votes)
At0:50, why did we go 4 points above?(4 votes)
Because the point is 4 above P, so we move the point up 4.(2 votes)
Would you have to understand 'squaring' numbers to firgure out this lesson?(3 votes)
Not necessary but it would be much easier if you understood.(3 votes)
- so we could multiply are first answer by 2?(4 votes)
- How do you know when to do the Pythagorean theorem on the triangle?(3 votes)
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!(3 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)
- are dilations only for scale factors(2 votes)
- If dilate a figure, you use a scale factor of how many times larger or smaller the figure is.
Like if segment AB is 2 units long, and you dilate that with a scale factor of 2, segment A'B' would be 4 units long.(1 vote)
1:06Video 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.