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Finding measures using rigid transformations

Rigid transformations preserve angles and distance. See how this behavior is used to find missing measures when given a triangle and the result of reflecting that triangle.

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  • starky sapling style avatar for user JF
    Mind if I ask what the Pythagorean Theorem is? I don't recall Khan Academy explaining it,
    so I'm a bit confused. Thanks for the help!
    (31 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      First, let's define a few terms. In a right triangle, the two sides forming the right angle are called legs, and the side opposite the right angle is called the hypotenuse.

      The Pythagorean Theorem states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. This is quite an important theorem in geometry!
      The theorem is usually expressed as the formula a^2 + b^2 = c^2, where a and b are the legs of the right triangle, and c is the hypotenuse of the right triangle.
      Note well: this does not mean that a + b = c.

      Example: if the legs of a right triangle are 3 and 4, then we can find the hypotenuse c by solving the equation 3^2 + 4^2 = c^2, which gives 9 + 16 = c^2, which gives 25 = c^2, which gives c = +-5. Since the hypotenuse c should not be negative, we discard c = -5. So the hypotenuse is c = 5.
      (2 votes)
  • starky sapling style avatar for user Daejien Beltran
    why do i not understand this
    (10 votes)
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  • leafers ultimate style avatar for user Huntertt90
    If there are four angles and I have to find one of the angles, and I add those three angles equal over 180, what do I do?
    (6 votes)
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    • mr pink green style avatar for user David Severin
      If a polynomial has 4 angles, it is a quadrilateral. As such, the angles inside of a quadrilateral add up to be 180(n - 2) where n is number of sides (or angles), so 180 (4-2) = 360. Thus, if you know 3 angles, add them together and subtract that total from 360.
      (10 votes)
  • piceratops ultimate style avatar for user Durgen
    a little tip for those who don't know, if a right triangle has 3(x) and 4(x), the length of the hypotenuse will always be 5(x) (x can be any real number). I figured this out long ago before I started using khan academy.
    (5 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      You're correct, of course. This is because you can prove the similarity of any two right triangles if they have two sides that are the same ratio apart and an included angle that is the same. Here, the right angle will be the same for every right triangle, and it is between the "3" side and the "4" side, so you can establish the similarity. From there, you can scale the sides and the hypotenuse however you want.
      Nice thinking!
      (7 votes)
  • leaf red style avatar for user John#yolo
    What is the Pythagorean Theorem?
    (5 votes)
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  • male robot hal style avatar for user Ethan thatcher
    why does he call the letters prime
    (5 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      In the video:
      ΔABC is reflected across line ℓ to form ΔA'B'C'

      ΔABC is read "triangle A B C"
      -and-
      ΔA'B'C' is read "triangle A-prime B-prime C-prime"

      See how the original triangle is called ΔABC, and the transformed triangle is called ΔA'B'C'?

      We keep the point names (A B & C) the same, so that we can easily see how each point from the original triangle (ΔABC) corresponds to the points of the transformed triangle (ΔA'B'C').

      And when we are speaking, we differentiate between the two triangles by using the suffix "prime" after each of the transformed triangle's points: for example, "A-prime B-prime C-prime". That way it is perfectly clear which triangle we are talking about.

      Also, when you look at the graph, you will know that the original triangle is the one without the prime markings (ΔABC), and the transformed triangle is the one with the prime markings (ΔA'B'C').

      Hope this helps!
      (5 votes)
  • starky seedling style avatar for user hridaya
    I have no clue how to doo this.
    (5 votes)
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    • old spice man blue style avatar for user Haritha
      its simple enough when you get the hang of it! let me walk you through it!
      when you use rigid transformations, such as reflections, lengths and angles do not change. using this, we know that the reflection of triangle ABC will have the same lengths and angles as triangle A'B'C'. then, you can use given measures to figure out the questions! i suggest watching the video again to see how Sal goes through it!
      hope this helped :)
      (5 votes)
  • duskpin ultimate style avatar for user agutu
    At , why did Sal refer to A'B'C' to 'a prime b prime c prime'? Is that how you pronounce that?
    (3 votes)
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  • orange juice squid orange style avatar for user Sunny Brightness
    were did he get the 90 from? :|
    (3 votes)
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  • purple pi purple style avatar for user louisaandgreta


    “You might immediately recognize that if you have a right triangle where one side is 3 and the other side is 4 that then the hypotenuse is 5”

    How you you immediately recognize that?
    Sal implies that the pythagorean theorem is not strictly necessary.
    (3 votes)
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    • mr pink green style avatar for user David Severin
      As you get further into Geometry, there are specific values that create what are called Pythagorean Triples, three sides of a triangle which will all be whole numbers. 3-4-5 is the simplest one, so people begin to recognize this pattern along with their multiples such as 6-8-10, 9-12-15, 12-16-20, etc.
      The next simplest is 5-12-13 then 8-15-17 and 7-24-25. People have calculated algorithms to find higher and higher ones but learning the first few are good for your math knowledge.
      (2 votes)

Video transcript

- [Instructor] We are told that triangle ABC, which is right over here, is reflected across line l, so it's reflected across line l right over here, to get to triangle A prime B prime C prime. Fair enough. So based on that, they're going to ask us some questions. And I encourage you to pause this video and see if you can figure out the answers to these questions on your own before I work through them. So the first question they say is well, what's A prime C prime? This is really what's the length of segment A prime C prime? So they want the length of this right over here. How do we figure that out? Well, the key realization here is a reflection is a rigid transformation. Rigid transformation, which is a very fancy word. But it's really just saying that it's a transformation where the length between corresponding points don't change. If we're talking about a shape like a triangle, the angle measures won't change, the perimeter won't change, and the area won't change. So we're gonna use the fact that the length between corresponding points won't change. So the length between A prime and C prime is gonna be the same as the length between A and C. So A prime C prime is going to be equal to AC, which is equal to they tell us right over there. That's this corresponding side of the triangle. That has a length of three. So we answered the first question. And maybe that gave you a good clue. And so I encourage you to keep pausing the video when you feel like you can have a go at it. Alright, the next question is what is the measure of angle B prime? So that's this angle right over here. And we're gonna use the exact same property. Angle B prime corresponds to angle B. It underwent a rigid transformation of a reflection. This would also be true if we had a translation, or if we had a rotation. And so right over here, the measure of angle B prime would be the same as the measure of angle B. But what is that going to be equal to? Well, we can use the fact that if we call that measure, let's just call that X. X plus 53 degrees, we'll do it all in degrees, plus 90 degrees, this right angle here. Well, the sum of the interior angles of a triangle add up to 180 degrees. And so what do we have? We could subtract, let's see, 53 plus 90 is X, plus 143 degrees is equal to 180 degrees. And so subtract 143 degrees from both sides. You will get X is equal to, let's see, 180 minus 40 would be 40. 80 minus 43 would be 37 degrees. X is equal to 37 degrees, so that is 37 degrees. If that's 37 degrees, then this is also going to be 37 degrees. Next, they ask us what is the area of triangle ABC? ABC. Well, it's gonna have the same area as A prime B prime C prime. And so a couple of ways we could think about it. We could try to find the area of A prime B prime C prime based on the fact that we already know that this length is three and this is a right triangle. Or we can use the fact that this length right over here, four, from A prime to B prime is gonna be the same thing as this length right over here, from A prime to B prime, which is four. And so the area of this triangle, especially this is a right triangle, it's quite straightforward, it's the base times the height times 1/2. So this area is gonna be 1/2 times the base, four, times the height, three, which is equal to 1/2 of 12, which is equal to six square units. And then last but not least, what's the perimeter of triangle A prime B prime C prime? Well, here we just used the Pythagorean Theorem to figure out the length of this hypotenuse. And we know that this is a length of three based on the whole rigid transformation and lengths are preserved. And so you might immediately recognize that if you have a right triangle where one side is three and the other side is four, that the hypotenuse is five. Three four five triangles. Or you could just the Pythagorean Theorem. You say three squared plus four squared, four squared is equal to let's just say the hypotenuse squared. Well, three squared plus four squared, that's nine plus 16. 25 is equal to the hypotenuse squared. And so the hypotenuse right over here will be equal to five. And so they're not asking us the length of the hypotenuse. They wanna know the perimeter. So it's gonna be four plus three plus five, which is equal to 12. The perimeter of either of those triangles, because it's just one's the image of the other under a rigid transformation. They're gonna have the same perimeter, the same area. The perimeter of either of the triangles is 12. The area of either of the triangles is six. And we're done.