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Show that if M is the midpoint of the line segment with endpoints (X1 Ya)(x2 Y2) then d(P M) + d(M,Q) =d(P Q) and d(P M)=d(MQ)To prove d(P M) + d(M,Q)=d(P Q}), firs...

Question

Show that if M is the midpoint of the line segment with endpoints (X1 Ya)(x2 Y2) then d(P M) + d(M,Q) =d(P Q) and d(P M)=d(MQ)To prove d(P M) + d(M,Q)=d(P Q}), first find the d(P Q). Suppose that = '(K1 Y1) &nd (*2 Y2) are two pointscoordinate plane , delermine d(P Q) using the distance formula Choose the correct ansiver below:d(P,Q) = { (*1-Y1)? (K2 - Y2)d(P Q) = 4 (*2 -*)r+ (Y2 -Y)rd(P,Q) = { (*1 -Y2)? - (*2 -

Show that if M is the midpoint of the line segment with endpoints (X1 Ya) (x2 Y2) then d(P M) + d(M,Q) =d(P Q) and d(P M)=d(MQ) To prove d(P M) + d(M,Q)=d(P Q}), first find the d(P Q). Suppose that = '(K1 Y1) &nd (*2 Y2) are two points coordinate plane , delermine d(P Q) using the distance formula Choose the correct ansiver below: d(P,Q) = { (*1-Y1)? (K2 - Y2) d(P Q) = 4 (*2 -*)r+ (Y2 -Y)r d(P,Q) = { (*1 -Y2)? - (*2 -



Answers

Prove that the midpoint $M$ of the line segment joining endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ has coordinates $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ by showing that the distance between $P$ and $M$ is equal to the distance between $M$ and $Q$ and that the sum of these distances is equal to the distance between $P$ and $Q$.

Okay, so we're discussing the midpoint formula, and we have that. If M is the midpoint of the line segment ending in points P of x one, y one and q of X two Y two, then the distance back from P M. Plus the distance of MQ is equal to the distance of peak, and the distance of Peter Em is equal to the distance of them to Q. So basically what we're saying is we have this line. M is here. P is here, and cue is here. So this distance needs to be equal to this distance. And if we add the distance from P T. M plus the distance from into Q, this is the distance from P to Q. Okay, so we use the midpoint formula to do this. So recall that the midpoint formula is equal to x one plus x to over to comma y one. Plus. Why, too, over to Okay, so now the distance from Pete M is equal to the square root of X one plus x two over to minus X one. So it's the whole distance minus X one squared, plus why one plus Why? To over to minus y one squared. It's the square root of all of this. Okay, so this simplifies to the square root of X one. Sorry, x two minus X one over to squared. Plus why? To minus y 1/2 squared. Okay, so now let's do the distance from M two Q. So this is equal to the square root of X two minus x one plus x two over to quantity squared. Plus why to minus why one plus y to over to quantity squared. So this reduces to the form X two minus X one squared over to so we see that this is equal to the same as P. M. And now if you add the distance from Peter M and the distance from Mt. Que, it will double. So the distance from P to Q is equal. Thio, the square root of X two minus X one squared plus why to minus y one squared. So we have shown what we were asked to show

In discussion. We have to prove that coordinates off eight point be. Thank you is an excellent, less extroverted. And why even less likely divided by you. So if any is X Y is the to make a point. Oh, lines like may Thank you, then distance between the and and is April who just stands Victor mean you and and that is X minus. Excellent. Police clashed less right minus Raglan. Well, just that it will do no extra minus X. Well, it's fair. Less y minus my line. All this threat No one was grating both sides. We will get X minus. Excellent. Holy scared Less right minus one. Despair extra, Linus. Excellent x. All is fair. That's why I minus violent, we'll square on simplifying. We'll get excellent minus extra. We're good lover. Excellent violence. Extra. Yes. Why? One minus right, my dear Lila, why do you want? Yes, right? Equals who? X multiply by excellent minus x 10 less Who? Emerging level like Linus. Why no on compare the coefficients of X one minus 69 11 minus weight on both sides. Big who works is equal to excellent, less extra. And who was is equal livened list by you. So X is equal. It'll excellent, less extra divided right room. And why is equal to liven us fight you might like, Which is the ordinary? Go made a point.

We have a problem number 36. And this we need to verify the mid-1 formula by showing that distance between the excellent by one and M. and the center of an M. And Q. Extra weight. We're both half of the resembled. Okay please. He is excellent. From AY. one. You is extra common. White too. An M. Bill hare. Midpoint. Excellent. Classics two x 2. And Why one Place? Why two x 2. So we have to verify the midpoint formula. Okay I'm so distance P. M. Will be equal to Excellent minus X one plus X. Two by two. Whole square. The survival minus Why one plus why to go to all square on the road? So this is the works one -X. 1. Excellent -X. two x 2. What's bad And why one Why two x 2 square in the room. This is the distance between B. And midpoint M. No Q. And midpoint M. Will have the distance. Ux X 2-. Excellent. Thanks to go to whole square plus Y tu minus of Y. one place By two x 2. Old square and the road. So this would be to extra minus X one. So simply X one minus X two by two Whole square place of arrival miles were two x 2. It's quite underwrote. Okay, so these two distance are the same. We can observe P. M. Equal to um Okay, no BQ distance will be equal to Excellent -X. two Whole Square plus why one minus Y. Two minus my tool square in the road. So if we define this by two, this will become excellent minus X. To buy whole square and one more thing we have to abide by to hear. So let us pay attention here and then qm. If we write Qm we can we can uh write it as excellent Sexual square. That's why one minus Y. Two. All squared by four under route. So this is excellent minus extra hole square. Why one of my Y. two old square. And the route they went by, too. So from here and here we can observe that you and we'll be able to B. Q x two. Hence, formula is very fired. And thank you.

To show that the midpoint of these two is that formula that's given. Well, first, we can find a midpoint. I m here. This is the same as first. The origin to pee and then 1/2 of p to Q. So, first, Opie, this is just the same as the vector. Uh, X one. Why one and then the vector P Q. This is going to be equal to. And then we're gonna have x two minus What are Sorry x two minus x one comma y tu minus Y one. So we're taking this plus 1/2 of this so it's going to be able to x one plus 1/2 of x two minus x one and then comma, and then we're going to have why to, uh, sorry. Ah, Why one plus 1/2 white to minus y one. Now, simplifying that here, we can see that I will have a 1/2 x two here and then on x one minus 1/2. X one is going to be a 1/2 x one. So we're gonna have 1/2 and then x one plus x two, and then next again, we're gonna have y one and then minus 1/2 blind one. So that's gonna be 1/2. Why one And when you have plus 1/2 y two. So that's gonna be the same as 1/2. Why one plus Why, too? Now, this vector can be the same as the point that is Midway, so we can put ah parentheses around to indicate that and is equal to this point here.


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