How to Construct Loci Equidistant from 2 intersecting Lines – ExplainingMaths com IGCSE GCSE Maths



welcome to this video about the loci equidistance from two lines we're going to construct it but before I do just a quick summary we've done loci from a points we've done loci from online loci equidistance to two points and now the final one equidistance to two intersecting lines and we realize that loci means all the points fitting a particular description okay so make sure you've seen those videos before you have a look at this one and you can check my site explaining maps those boom well you're gonna find out my resources nicely put together for you okay now let's say I have the line a B okay over there and I'll make another line let's say of an angle of 60 degrees AC okay now that this could be the shore perhaps of so this is the land okay and then we have the C in the middle and this boat wants to make sure that it sells exactly equidistance of a C to a B so exactly in the middle yeah so just your gut feeling exactly equal is from those two lines you know this boat has to roughly seal like that do we agree with that equidistance yeah so I'm not going to silver here I'm not gonna seal here because that's near its baby in a to AC now I'm gonna seal exactly in the middle but this is not estimation yeah this is not science guys this has maps yeah we do things properly and accurately so how do we do that we have to construct the angle bisector well done yeah so before I do very quickly loci from a point you do the circle okay with a radius looking for particular amounts loci from a line parallel lines yeah but then at the vertices semi circles depending on the question anyway this is from two points in my previous video the perpendicular bisector or line bisector an equidistant from two lines it is the angle bisector now I told you this is an angle of 60 degrees it's going to be 30 30 degrees but we have to strucked it and in other videos I've shown you how to do that but I'll show you now again it's very easy you have a compass and make sure you buy a compress they're not expensive but when you open it that it stays in the same position sorry excuse me yeah so that it doesn't move okay make sure you buy a proper compass okay what do I do I open my compass doesn't matter how far but not too far because it's difficult to work with not too close possessed it puts work if I just open it a couple of centimeters but once I decide upon a particular wheel doesn't matter how wide once I've decided I gotta stay or stick to that way okay what am I going to do I'm gonna put the needle in my angle and I'm gonna intersects intersects I should say those legs that creates my my angle okay so that's there and there I'm not sure if you can see that so I'm in black I'm just going to just going to trace that with Marco so this art of the server but are okay I do not have to draw the entire circle the other this is going to confuse me I'm only interested in those intersecting parts with the arms of my angle okay very carefully I'm not going to change the width and if I accidentally change the with my compass I just have to start over again doesn't matter but if the width is still say I'm going to put my needle now at this point of intersection and draw this arc of the circle and I'm gonna do the same from this point of intersection and roll that Park over there now why again don't I not do I don't have to draw the entire circle I know that my angular bisector is going in that direction so I'm looking for this point of intersection and again I'll trace what I just did with my compass I'm looking for that point of intersection over here yeah so I don't have to draw the the rest of the circle because from my angle through that point of intersection so my estimation early on was quite accurate actually this is my Oh angular bisector and I'll tell you that a long time ago I was in the Navy and we used to do this all the time yeah creating the loci of points equidistant to two line set because that we knew how to navigate the ship and I'm pretty sure that's still doings okay so it's the angular bisector good like I can't share if this was useful guys then I can help your friends to check my site explaining my hostel Tacoma you can ask me questions there if you want in a forum I'll help you you'll find all the videos basically you need to pass your Maps ourselves yeah all for free by the way I'll see you it

How to construct angle of 90 degree using Compass



welcome to education channel this video I am going to explain you how to consider to nineteen ninety degrees using only compass how to construct 90 degrees angle with compass and explain first we'll take a line segment now at this point at this point I want exactly consider to a 90 degrees angle so what you take first this from this point any point this is an American construct 90 degrees so I'm going to concerts night right to say I mean 90 degrees line there's some reasonable radius so don't take more than a very small take some radius and this is a center point and drawn out actually 90 is off of the 180 right 180 divided by two is ninety if I can find out 180 degree point then I can buy 6 0 and 180 from here I draw a not without changing the radius previous how much until you drop how much radius you are drawn without changing this now me here cut this arc and again from this point became Amarok again from here one more on that means this is 60 degrees and this is 120 degrees and remaining is 180 degree now this is a zero point and this is 180 degree point right now 0 and 185 bisect then 90 will come need not to extend the line at this point I want know this is the a so at the point yeah I mean 90 degrees so from a troy log first 60 120 and 180 three arcs if I got now from 0 and this one zero and 180 I have to bisect so from zero just take more radius draw an arc a bow because I am visible again same radius from 180 mark because I'm bisecting zero and 180 so 90 we come the simple logic 0 arc and from our nadh on dry not with the same radius no wait they join the both points just to draw a line from the now this angle if you keep the point C so angle B is exactly 90 degrees so you can check with protractor I kept it so exactly 90 degrees so this is the way we can construct until 90 degrees at any point okay we'll check from this is right curves one more thing I'll show you one more example I would construct 90 degrees they said awesome so what we'll take from the center point we just draw an arc right with the same radius factor to us and from yeah so zero point and one negative point in extent zero and 180 so I need to bisect this now this is 180 point cut our kaboom and from here be same they both both six from the zero point and 180 point and then joy if we join the two points so this is the angle 90 so you can use any side any way to not only these any line segment is given at any point we can construct 90 degrees using compass

(I.46) Construct a square from a given line, Proof



in proposition 46 we proved that with any given line we can always construct a square to begin we are given line a be first with point a and line a B we apply proposition 11 to construct a perpendicular line AC therefore angle BAC is a right angle next with line a B and line AC we apply proposition 3 to make line ad congruent to line a eb now with line a b and point d we apply proposition 31 to construct a parallel line through point D therefore line de is parallel to line a be next with line ad and point B we apply proposition 31 again to construct a parallel line through point B therefore line ve is parallel to line ad since we have two pairs of parallel lines that create shape a de be by definition 6a de be is a parallelogram now since a DB is a parallelogram by proposition 34 the opposite sides are congruent therefore line a B is congruent to line de and line b e is congruent to line ad so now we have line ad is congruent to line a B and line a B is congruent to line de with these two facts we can apply axiom one to show that all three lines are congruent since we also have the line b e is congruent to line ad then by applying axiom one we can show that all four lines are congruent next looking at the parallel lines de and a B we can see that both lines are intersected by line ad then by proposition 29 the sum of the two interior angles on the same side are congruent to two right angles therefore angle B ad plus angle a de are congruent to the sum of two right angles now angle BAC is a right angle which can also be called angle B ad this means that angle a de must also be a right angle since angle B ad plus angle a de are congruent to the sum of two right angles now since a de be is a parallelogram by proposition 34 the opposite angles are congruent therefore angle B ad is congruent to angle de B and angle a de is congruent to angle a be e so by axiom one we can show that all angles are right angles now since a de be is a parallelogram that has all of its sides congruent and all the interior angles are right angles then by definition 6a de b is a square therefore we have proven that with any given line we can always construct a square you

Construct Rectangle.m4v



okay let's construct a rectangle where the short legs will be this length of a B and the long legs will be this length of CD the first thing I need to do of course is take my straightedge give myself a nice long segment to work from and let's go ahead and start with the CD I like long rectangles rather than tall that's just my preference so first of all there see I'm going to come up and measure with my compass the length of CD there's my construction work I'm going to come down here to see make that construction work and I now know that that is point d well now what I need to do is draw the perpendiculars with my compass so I need to start by giving myself points that are equidistant to see okay so there's two points that are equidistant and we'll start on this side open my compass up and i'm going to start what i like to call the cats i come up now i'm going over to that other point which was equidistant finish off the cat side and now i know if i use my pencil and my straight edge through these two points of intersections that i will create a perpendicular through that point c make it nice and long all right i need to do the same thing over here from those two points of intersection i now need to create my cat's I so I open the compass up past the point D and I get that nice smooth art I'm going to come over to my other point equidistant make that nice smooth arc and again using my pencil on my straight edge I get a perpendicular through d now remember make it nice and long ok so now I have a right angle here and a right angle here my rectangle is very close to being complete but here's my long edge CD now I need to create a be so I need to bring my compass back up and I'm going to measure with my compass a be show my construction work now from see I'm going to make that point and I'm going to label it right there this could be like a bee and keeping the same setting on my compass I'm gonna come over here to the D put the mark and I know that right here this is also a B and these two are now congruent to each other and finally I can take my straightedge connect these points and because these are right angles these should be right angles and because this is CD this should also be CD and I can always check that if I'm not sure by coming down and measuring my CD with my compass and then coming up here from a to a-put bad names that you get the idea and sure enough it's the same length so now these two are congruent to each other and I've created a rectangle

Constructing an Altitude of a Triangle



welcome to a lesson on how to construct an altitude of a triangle an altitude is a line segment from a vertex that is perpendicular to the opposite side so this red segment is an altitude because it's a line segment from this vertex to the opposite side and is perpendicular to the opposite side each triangle has three altitudes from each of our techs and the point of concurrency is called the orthocenter so here 0.0 is the point of concurrency with a point of intersection of the three altitudes and this is called the orthocenter now we're going to take a look at constructing an altitude for an acute triangle in an up to triangle and for these constructions we're going to need a compass and a straightedge so for this example we have an acute triangle and we'll construct the altitude from this vertex so the first step is to put the point of the compass on this vertex and open it far enough so that when we swing an arc it will intersect the opposite side in two points and if you want you can extend this side of the triangle so in this example if we swing an arc that look similar to this notice how it intersects the opposite side in two points we have a point of intersection here and we have another point of intersection here now we're going to take the compass and put the point on one of the points of intersection and then open it so that it's more than half the distance from this intersection point to this intersection point and once we do that we'll swing an arc above and below this side of the triangle so it might look something like this we want to make sure we swing them so that they would extend past what we think would be the midpoint of the segment between these two points of intersection and now without adjusting the compass we'll put the point on the other point of intersection here and then swing another arc above and below this side of the triangle and those arcs might look something like this now we're going to take our straightedge and line up the intersection of the arcs here and here with this vertex and this should be our altitude so this red segment would be our altitude because it connects this vertex to the opposite side and is perpendicular to that opposite side now let's take a look at the construction when we have an obtuse triangle if we want to construct the altitude from this vertex here to the opposite side notice how we're going to have to extend the opposite side in order to make a segment that's perpendicular to that side so with our straightedge we'll go ahead and extend this side and then we'll perform the same procedure we'll take the point of our compass put it here on the vertex and then we'll open up the compass large enough so that when we swing an arc it will intersect the opposite side in two points and when we swing an arc it might look something like this and notice how it intersects the opposite side in two points we have one point of intersection here and one point of intersection here next we'll take the compass and put the point on one of the points of intersection and then open up the compass so that extends past half the distance from these two points of intersection and then swing an arc above and below this side of the triangle so if we put the point on this point of intersection and swing an arc above and below it might look something like this and then without adjusting the compass we'll put the point on the other point of intersection and do the same swing an arc above and below the side of the triangle so it might look something like this now if we line up the points of intersection of these arcs with this vertex it should form our altitude so with a straight edge we'll line these three points up they should be linear and construct our altitude and notice how this segment is perpendicular to the opposite side here therefore this should be our altitude I hope you found this helpful you