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## High school geometry

### Course: High school geometry>Unit 2

Lesson 1: Rigid transformations overview

# Finding measures using rigid transformations

Finding measures after a rigid transformation, like a reflection, is pretty simple! Since the shape and size stay the same, the lengths of corresponding sides and angle measures remain unchanged. Area and perimeter depend on the side lengths, so they stay the same too. So, if we know the measures of the original figure, we can use those same measures for the transformed figure.

## 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.
• why do i not understand this
• Start from the fundamentals(very basics) and then go on higher level.
• 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?
• 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.
• 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!
• What is the Pythagorean Theorem?
• The Pythagorean Theorem is the formula:

a^2 + b^2 = c^2

Where a and b are the legs of a right triangle and c is the hypotenuse of that triangle.
• why does he call the letters prime
• 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!
• I have no clue how to doo this.
• 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?
• Yes. The " ' " means "prime" or " 's image" in the context of "A's image"
• I didn't get how he found the measure with only one number available? And how do i do this on a triangle that most tests have that has no rights angles?

All the angles of a triangle will always add up to 180°. The measure of angle A is a right angle (90°), and the measure of angle C is 53°. Therefore, angle B is (180-90-53), which is 37°, and as angle B and angle B' are the same, the measure of angle B' is 37°.