Yahya is attempting to prove the cosine rule for the...

Yahya is attempting to prove the cosine rule for the triangle shown below. He starts by adding the dotted lines as shown. Here are the first few lines of his proof. y² = c² – (b + x)² [Pythagoras's Theorem] y² = a² – x² [Pythagoras's Theorem] c² – b² – 2bx = a² [Equating expressions for y² and tidying up] Complete his proof explaining clearly what you have done.

Transcript

00:01 So for this question, we're trying to prove the rule of cosines or the law cosines.

00:06 And we're seeing here, a drawing is given.

00:10 And the first couple steps are also given.

00:12 So we're given that.

00:14 So let me just kind of highlight those steps for you here.

00:17 So if i look at this big triangle here, sorry about that.

00:22 If i look at this big triangle here, okay, this is the triangle i'm going to be focusing on when i write this first one.

00:29 So just keep that in mind.

00:30 I'm going to erase, get rid of that because it will get too crowded.

00:33 If i don't.

00:34 And so i'm going to have y squared equals c squared minus b plus x squared.

00:46 And that's given to us by pythagoras theorem.

00:53 Okay.

00:53 So you can see there my y squared and then b plus x squared on the bottom is going to equal that c squared.

00:59 So they just subtracted the quantity b plus x quantity squared from both sides.

01:03 And then this time focusing on the smaller right triangle.

01:08 We can see there with legs y and x in hypotenuse a we get y squared equals a squared minus x squared and again this is by pythagorean theorem okay and then the next step is just to equate these two things so they put them equal to each other i'm going to go ahead and do a step out here that they didn't do i'll go ahead and say here's the left hand side this is c squared minus when i square this, i'm going to get b squared minus 2bx, sorry, plus 2bx, plus x squared.

01:56 And i'm going to equate that to a squared minus x squared.

02:02 Okay, so i just did, i just squared the, i squared this term to get this term.

02:08 All right.

02:09 And then it's going to give me c squared minus b squared minus 2bx.

02:17 Minus x squared equals a squared minus x squared and then we can see here's a minus x squared over here here's a minus x squared over here i forgot to write the squared and then those can be uh dropped off both sides okay so that's where they get the third step here of cleaning up a bit right this is equating and tidying of a bit so that's what they tidied up okay and so keep in mind here that i'm trying to prove law cosines and i have this angle c right here.

02:54 I have this angle c right here.

02:56 So really what i want to solve for is this side, right? so c squared.

03:02 So if i solve this now for c squared, i'm going to get c squared equals a squared.

03:11 I'll be adding b squared to both sides, plus b squared.

03:15 I'll be adding 2bx to both sides to bx...

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