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Let E be the point _ al which the bisector of LABC intersects Drop a perpendicular from E to AB and call the foot F....

Question

Let E be the point _ al which the bisector of LABC intersects Drop a perpendicular from E to AB and call the foot F.

Let E be the point _ al which the bisector of LABC intersects Drop a perpendicular from E to AB and call the foot F.



Answers

$\overline{\mathrm{AB}}$ is a chord of circle $\mathrm{E},$ and $\mathrm{C}$ is the midpoint of $\overline{\mathrm{AB}}$. Prove that $\overrightarrow{\mathrm{EC}}$ is the perpendicular bisector of chord $\overline{\mathrm{AB}}$.

This problem asks us to show that the line Y equals X is a purple dickler by sector of and points A B and B A where I cannot equal b Um, so let's check this out. First of all, the slope of y equals X is equal to M equals one. The slope is one meaning that the slope for something perpendicular to that should be negative one. So let's double check. So I'm using the slope formula here. M y tu minus y one over extra minus X one. I labeled my A B B A coordinates at the top wrote it in a minus. B over B minus. A is equal to What I can do is factor out a negative one from the top, and I'll end up with negative A plus B over B minus a, which are equivalent meaning I can cancel those out so you get a slope of negative one, which is the inverse and negative of positive one. So these two values are perpendicular to each other, so that checks off one of the one of the conditions here. We're trying to look for perpendicular, and now the other point is by sectors. We're going to look at the midpoint form. Midpoint formula is X one plus x 2/2 with the Y one plus y to over to, um, as the other coordinate. So let's check this out as in terms of A and B so X one would be a plus b over to um, and then y one y two would be B plus a over two. Now addition. We can switch the order of those around to make it a plus B over to, and what you'll notice here for our X and Y coordinate is that X is equal to y A plus B over two. A plus B over two means that it would fall on the line X equals y or y equals X. Therefore, it by sex. It gets bisected by the line y equals X, so that proves the to the perpendicular. By sector Y equals X for the points A B B A

For Problem number 19 were given a segment E f and asked to find the perpendicular by sector. Er, excuse me. The equation for the particular by sector of this segment. So first, I'm gonna just graft my segments so that I can visually see it better. So now that I have my segment, I know I have to find the perpendicular by sector of this segment. So, first of all, perpendicular by sectors, um, they a line perpendicular early, which means they create the right angle and they bisect, which means they cut the segment in half. So there's two things I have to do in order to create this equation, especially because I'm trying to create the equation and point slope form. So the first thing I want to do is find the mid point of the segment e f because the midpoint is gonna be where this segment is cut in half. So I'm gonna use the midpoint formula to find the midpoint of segment E f. I went ahead and labeled my points x one y one x two y two because it helps me place these into the formula. Better also, you always start out with the 1st 1 I give you as X one and white one. So now that I have labeled and the formula set up, I can simply plug them in. So x one is negative for in X 20 divide both of those by two. Why one is negative seven and why two is one. And I divide but those by two. So I'm adding on both of these and this is set up like a coordinate points. So when I simplify it, it's going to come out into an exact point and you can simplify this different ways. Some people like to just simplify the X all the way down, and some people just then simplify the why I just go ahead and do it all at the same time. So negative four plus zero is negative. Four and negative seven plus one is negative. Six. Now the last thing you have to do is divide both by two so negative four divided by two negative two and negative six divided by two is negative. Three. So my mid point is the point negative to negative three. So now that I have that plotted, I can see that it cuts the second half. Of course, my segment is not drawn the perfect scale, so it doesn't look super in half. But it is so now that I have the midpoint, I know that that point lies on both e f and the perpendicular by sector, because the perpendicular by sector runs through that point for particularly. So how do I find the line that's perpendicular to segment E F. Well, I know that perpendicular lines have perpendicular slopes, So if I confined what the slope of e f is, then I can change that into its perpendicular slope to get the slope of the perpendicular by sector. So I need to use the slope formula. So I have my slip formula written, and I still have my labels from before. So now I have to do is plug in the different coordinate. So why two is one so one minus? What I want is a negative seven. I put the negatives in parentheses over X two is zero minus. X one is negative for and again over the negative inferences because I am supposed to be subtracting. So now I just simplify. Whenever I see two negatives, I know that that creates a positive. So one minus 97 is the same as one plus seven, which is eight and then zero minus negative. Four is the same thing. A zero plus four, which is just four, then eight divided by four is too. So the slope of E f is two or 2/1. Well, I want the perpendicular right sector. So So I need the perpendicular slope to tube. So perpendicular slopes are opposite reciprocal. So to make this the opposite, I see that it's positive. So the opposite of positive is a negative. So my perpendicular slopes gonna be negative, which we know because it's going down and then reciprocal means I flipped this upside down. So what's on the bottom now goes on the top and what's on the top now goes on the bottom. So the slope of my perpendicular by sector is negative 1/2. So what they're asking out of me is to find the equation in point slope form of the perpendicular by sector. Well, I have a point on the perpendicular by sector, and I have the slope of the perpendicular by sector, so all I have to do is plug it into point slope form. So now I have point Slope form written and I labeled my new point x one y one because this is the equation of a different line for the perpendicular by sector. So now I just plug in. Why, minus why one which is negative. Three equals my slope, which is negative 1/2 times x minus X one, which is negative. Two. So I'm almost finished. Are just need to simplify this expression because it's not mathematically correct to have two negatives with parentheses, so I know when I see that it really turns into a positive. So my final answer is why plus three equals negative 1/2 times X plus two and that is correct. Point slope form of the perpendicular by sector of the segment E F

So if we know that a B is a perp into goodbye Sector X Y, let's struck there. So yeah, X, Why do we want a B to B A perpendicular by sector, but not the other way around. So if you guys know of the situation of a square is a rectangle, but a rectangle, it's not square. It's kind of like that. I'm gonna draw it where it's like this where this is a B so you can see that a B is perpendicular and biceps.

Going to do problem number 21 this question What we have to do? We have to judge toe prove that C is the perpendicular by sector off baby. So the coordinator is given that is minus four comma minus three for coordinator be is given that it's six comma, one on coordinate off See is given that this five comma minus 11. If she is the par particular by sector, then a C sort of equals toe BC So let us find the distance off a C A. C is We can find a C by the formula that is for extra many sex when all the sculptures were to Venezuelan Holy square. So five Manus Manus for that is nine holy square plus Why to Venezuelan that is minus 11. Menace Ministry that is minus 11 plus three. So that will be equals two Eat minus 11 minus 11 plus three. So that'll be calls to Manus H Holy square. Okay, so what we're getting here? We're getting a nine square. There is 81 plus 64. So this is equals toe four plus one is five and eight plus six is 14. Similarly, we're going to find for B C. Let's see what it comes NBC Again. We're going to use the same formula that extra money. Sex, unholy square place. Where to? Venezuelan? Holy square. So here, X two is again five but X minus six. So five minutes sixties man is one Holy square. Plus again this really ministerial square. So we're going to get here that it's 145. Okay, so from this we got that S e Is it calls too busy? Yes, he is. It called CBC. So it has proved that since the perpendicular by sector off lines negatively, there's some


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