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Use the diagram of $\Delta \mathrm{ABC}$ Copy and complete the proof of the PythagoreanTheorem (Theorem 9.1$)$\begin{array}{cl}{\text { Given }} & {\text { In } \Delta \mathrm{ABC}_{5} \angle \mathrm{BCA}} \\ {} & {\text { is a right angle. }}\end{array}Prove $c^{2}=a^{2}=b^{2}$

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Geometry

Chapter 9

Right Triangles and Trigonometry

Section 3

Similar Right Triangles

Right Triangles

Circles

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so this problem were given this poorly drawn figure by me. But it's an exact copy. Truly some extent of the figure from the textbook. Well, we're giving this figure, and we're giving a series of statements and reasons with some blanks in them. And our goal is to construct a proof here such that we have C Square using a script was B school here proves that by tag. Rinse there. And as you can probably tell, there are many, many, many ways to prove the pathetic in Durham. And this is just one that the waste we could do it. So the first statements here say, Is that this triangle, ABC here, um, this is an angle B. C. A. Is a right angle here, and for that reason for a number one, I'll put that as our one that is just given to us because we're shown here. That is a right angle, Um, because it's a little square Books, either. Now, step two here is to drop her particular segment or the altitude from sea to line a beef. That's why the perpendicular postulates. It's basically you could just draw a multitude there. I'm really does know. What are you doing? Pure reason for Just draw awesome line segments such that this is true There, that's what we have now the next four year is, um for our three, it's a reason. Three. It says that C e is equal to a squared and C f is equal to be square there. Right? So in other words here, however, we have actually let me put this Steven three. So they seem in three here has C e length seat. So, um, Val, you see which is like a bee Times e is equal to a squirt, which is bc sward this word and cf is he going to be swear it there to cf here, which is a B All right. Time's F, which is a d a b times a d is equal to be score a sea squirt there for that's here. That would be the geometric meet there. Right, So geometric mean fair. Um, there. Hi. That's a term that we use. Drop this chapter there. And that's how it's set up now for statement for, um we are given that CEO here actually, let me you underlying the answers I were given that C is equal to eight. So I see. Plus B squared is equal to blink, which puts a green underlined, um, plus B squared there. Well, we know that C B c plus B squared was blank, crispy square there. But that means that we can do some substitution and realize that won't see easy with a squared. So that means a C plus B squared. Here's what's on the left hand side and I'd better have a scruples be scored on the right hand site. That's a addition. Property of quality there. Now substituting here, um, for a reason. Five. I'm actually press statement five our semen five here They say that c e plus C f is he with a squared plus B squared there. Um, in that case, for reason. Five there, um, you could either put addition profit of equality or what I think is more appropriate. It's just by substitution there. Since you know that CFC with the B squared, Then when I replace it, be school to see up there. Um, next thing they do here is the factor out sees a Sikh was e plus f there, which is a squared plus B squared there for reason. Six. Um, that would be the distributive property. Yeah, I hopefully that orb public by factoring were actually applying the reverse of the distributive property there. Um, now for a statement seven. We know that people's f then hair is equal to something. But by looking at the fear he plans f here is equal to see So And the reason seven they give you here is by the segment dish impartial it there. Okay. And well, since we know that plus a 50 will see here, then by substituting your we get see, Time C is equal to a squared plus B squared there for that reason. Oh, we know that you pull some b school, see somethin substitute back into the equation we have for Stephen six. So reason eight. Then we'll just be substitution. And then, um, the proof finishes off by simplifying care. I'm with the ninth statement in which c squared is you go to a sward. Those pieces were there that reason just being simplify there. Okay. So anything underlined in green hair is the answer, and not that it's home. Pretty basic. There wasn't be commonly used in algebra and geometry there. I, um we have the geometric mean term, which is the emphasis of this section. Um, and we just in substitution here and fact twosome factoring there. So factor here to manipulate the alleged room. Such that, um, I get by by ground zero.

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