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Course: 8th grade (Illustrative Mathematics) > Unit 1
Lesson 4: Lesson 7: No bending or stretchingFinding 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.
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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!(44 votes)
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.(66 votes)
- why do i not understand this(12 votes)
Start from the fundamentals(very basics) and then go on higher level.(15 votes)
What is the Pythagorean Theorem?(6 votes)
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.(11 votes)
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)
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)
I have no clue how to doo this.(5 votes)
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 :)(6 votes)
why does he call the letters prime(5 votes)- 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)
- I jus answered all he questions before the video started.(5 votes)
- At3:06, why did Sal refer to A'B'C' to 'a prime b prime c prime'? Is that how you pronounce that?(2 votes)
- Yes. The " ' " means "prime" or " 's image" in the context of "A's image"(5 votes)
im just kinda confused on this unit.(3 votes)- I'm glad this is all starting to click and make sense.(4 votes)
3:06Video 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.