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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.

## Want to join the conversation?

• 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! • 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.
• • 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? • 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. • 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!
• • • 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!
• • 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 :)
• At , why did Sal refer to A'B'C' to 'a prime b prime c prime'? Is that how you pronounce that? • • “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. 