1 00:00:01,139 --> 00:00:09,820 we are going to solve the tangencies worksheet let's start with exercise number one so we are 2 00:00:09,820 --> 00:00:17,579 demanded to draw a tangent line to a circumference of radius 27. first we are going to draw a line 3 00:00:17,579 --> 00:00:25,660 in any direction and there is where we are going to measure the radius okay so with a ruler 4 00:00:25,660 --> 00:00:47,729 we are going to measure 27 a clean line in 27 the arrowhead inside let's draw the arrowhead 5 00:00:47,729 --> 00:01:01,649 clean and with the tip ending on 27 exactly okay this is radius 27 6 00:01:01,649 --> 00:01:19,379 27 we erased the rest of the line leaving the mark there and we are going to draw sorry 7 00:01:19,379 --> 00:01:29,480 we are going to draw the line 27 the circumference radius 27 sorry 8 00:01:29,480 --> 00:01:50,480 oh sorry about that okay so once we have that this circumference is a little bit dirty over 9 00:01:50,480 --> 00:01:59,959 there okay my compass is not a very good one okay excuses but um now we are going to place 10 00:01:59,959 --> 00:02:08,439 the tangency point whatever you want okay so we can choose for example tangency point here 11 00:02:08,439 --> 00:02:18,800 an empty circle like that one this is going to be t the tangency point and for solving the 12 00:02:18,800 --> 00:02:31,960 tangency you have to connect o with t that's another radius and the tangent needs to be 13 00:02:31,960 --> 00:02:48,520 perpendicular to that radius so you can place the square like this or the bevel and the other one 14 00:02:48,520 --> 00:02:59,479 like that move it a little bit keeping the right angle and moving it a little bit down so the point 15 00:02:59,479 --> 00:03:12,939 is free and now we can draw the tangent both sides that's a tangent okay we are going to 16 00:03:12,939 --> 00:03:24,939 name this that tangent as line t lowercase and use the perpendicular symbol with a little dot 17 00:03:24,939 --> 00:03:37,069 inside little arc with a little dot inside first is exercise two and we will draw first 18 00:03:37,069 --> 00:03:44,849 30 millimeters radius circumference on center o okay so with the ruler we will measure 19 00:03:44,849 --> 00:04:04,349 30 millimeters like that okay and we draw the circumference okay that the circumference 20 00:04:04,349 --> 00:04:17,750 radius 30 we normally we don't use those lines to name the radius but okay it's clear enough 21 00:04:17,750 --> 00:04:30,389 now we need we are demanded to do an external r18 circumference on r r is this line okay 22 00:04:30,389 --> 00:04:44,470 that's r so we are going to take this intersection point as a tangency point this is t okay and the 23 00:04:44,470 --> 00:04:59,959 right the radius should be or must be 18 so you will measure 18 from t point down so here 18 24 00:04:59,959 --> 00:05:15,319 that point will be the solution to the first part this is going to be o1 and we are going to call 25 00:05:15,319 --> 00:05:32,920 this tangency point t12 so they are named alike so this is radius radius 18 and we are going to draw 26 00:05:32,920 --> 00:05:49,310 the first solution you place the compass on 01 like this okay that's 01 and you draw the solution 27 00:05:49,310 --> 00:05:56,290 this this shouldn't cross each other okay they only contact in one point 28 00:05:56,290 --> 00:06:06,269 this is not clean enough okay and now um on the other part of the exercise we are demanded to 29 00:06:06,269 --> 00:06:17,649 solve an r23 internal circumference on s s is this one okay this line so we are going to measure 30 00:06:17,649 --> 00:06:43,420 23 this maybe naming r30 here wasn't so clever okay so i'm going to erase that i'm going to name 31 00:06:43,420 --> 00:06:58,000 this o2 and this line this point sorry sorry that point is going to be c2 32 00:06:58,000 --> 00:07:14,699 um r30 we are going to name here for example this is going to be a radius you can do this 33 00:07:14,699 --> 00:07:33,449 in any direction too so this is going to be r30 okay and um and now we are going to name this as 34 00:07:33,449 --> 00:07:51,269 r23 okay sorry for the mess here because this is not so clean okay so r23 r30 if you now place 35 00:07:51,269 --> 00:08:14,069 compass on o2 like this and take the radius o2 t2 you will draw an internal sorry internal 36 00:08:15,110 --> 00:08:31,959 tangent circumference like okay excuse me okay sorry about that because i having some problems 37 00:08:31,959 --> 00:08:40,519 with uh with the compass and with the light so you can see here this is really a mess okay this 38 00:08:41,559 --> 00:08:51,399 must not happen never okay so it should be clean like that one okay so from o2 to t2 39 00:08:52,840 --> 00:09:00,360 second solution from o1 to t1 first solution okay remember the measurements be precise 40 00:09:01,960 --> 00:09:31,799 Exercise number three. Tangent circumferences are 18 and 30 to the line on tangency points. Okay. To the line. So the infinite circumferences that could be tangent to this line will have their centers on this perpendicular. 41 00:09:31,799 --> 00:09:40,700 so if you place if you you place the first ruler in a right angle and the second ruler like this 42 00:09:40,700 --> 00:09:49,960 you take this out and you draw the center's line okay so that's a perpendicular line this is a 43 00:09:49,960 --> 00:10:01,649 right angle okay we are going to name the right angle okay sorry about the light again 44 00:10:01,649 --> 00:10:12,960 this is the right angle here and now this is really really easy because we have to measure 45 00:10:12,960 --> 00:10:33,639 18 i'm going to zoom it a little bit so 18 here and 30 on top be precise like always 46 00:10:33,639 --> 00:10:48,960 last set so this is going to be r18 in there r18 in this is going to be 47 00:10:48,960 --> 00:11:00,860 r30 first solution o1 second solution o2 it's true that you could find 48 00:11:00,860 --> 00:11:09,200 two more solutions so you can draw a 18 circle to the other side a 30 circle to the other side 49 00:11:09,200 --> 00:11:17,320 radius but we are going to give one solution one solution is enough okay so great now we are going 50 00:11:17,320 --> 00:11:27,860 to solve it so placing the compass on o2 till the tangency point t we draw the circumference 51 00:11:27,860 --> 00:11:39,759 okay our solution and we do the same with o1 center till point c again 52 00:11:39,759 --> 00:11:54,450 tangency point and we draw the second solution okay if in class you did the other way around 53 00:11:54,450 --> 00:12:06,549 so if you draw the 13 radius below it's okay and the 18 radius on top it's okay okay it's good 54 00:12:06,549 --> 00:12:14,970 enough great and now we are going to solve number four in number four you have to draw 55 00:12:14,970 --> 00:12:26,730 15 and 28 tangent circumferences to both lines so this is an angle okay r and s great so for 56 00:12:26,730 --> 00:12:37,610 for placing these solutions we need first we need to draw um an angle bisector for drawing 57 00:12:37,610 --> 00:12:48,789 an angle bisector you need to we are going to name this point as v okay this is going to be 58 00:12:48,789 --> 00:13:00,470 the vertex of the angle that's b and for drawing the angle bisector you need to draw one radius 59 00:13:00,470 --> 00:13:12,120 from b not too big not not too small okay something like that with any radius not too big 60 00:13:12,120 --> 00:13:26,620 not too small um and we name we name those two points as one and two okay i'm going to zoom it 61 00:13:26,620 --> 00:13:38,539 little bit so with any radius that arc okay you cross both lines and now from one and two you're 62 00:13:38,539 --> 00:13:47,740 going to do two arches okay with any radius again you can change the radius from the first one okay 63 00:13:48,299 --> 00:13:54,059 so we are going to take for example this one any radius again you choose the radius 64 00:13:54,059 --> 00:14:05,830 you do that arc draw that arc and from two you do the same okay again that's from one 65 00:14:05,830 --> 00:14:15,190 with any radius and with the same radius from two so the arches cross each other now we are going to 66 00:14:15,190 --> 00:14:27,610 place exactly sorry for the light to place exactly the crossing point okay and connect 67 00:14:27,610 --> 00:14:40,129 this is going to be named three and we are going to connect i'm going to zoom out v with three 68 00:14:40,129 --> 00:14:47,389 okay and by doing that i'm going to change the ruler by doing that 69 00:14:47,389 --> 00:14:59,289 vm3 you will get the set the the angle sorry bisector okay if you do this and these two parts 70 00:14:59,289 --> 00:15:08,950 doesn't seem equal repeat okay because angle bisector sometimes are tricky okay and now 71 00:15:09,909 --> 00:15:17,350 here on that angle by sector we would have infinite circumferences tangent to the to both 72 00:15:17,350 --> 00:15:29,029 lines but we only need radius 15 and radius 28 tangents so for placing those two solutions we 73 00:15:29,029 --> 00:15:38,309 need to draw parallel lines to draw that parallel lines i'm going to zoom it a little bit 74 00:15:40,519 --> 00:15:58,379 to draw that parallel line we are going to need perpendicular okay so here 75 00:15:58,379 --> 00:16:13,759 perpendicular to S as far as you can go okay so here on S as right as you can go 76 00:16:13,759 --> 00:16:28,769 draw a perpendicular here okay I'm more or less one centimeter apart something 77 00:16:28,769 --> 00:16:38,899 like that one centimeter apart we are going to draw another perpendicular if 78 00:16:38,899 --> 00:16:45,720 you want to do it exactly you can also use this ruler here on line s and this 79 00:16:45,720 --> 00:17:01,679 other one on top okay good so doing that you will get the other perpendicular okay so right 80 00:17:01,679 --> 00:17:12,160 angle here and right angle here on both and now we are going to measure the 28 and the 15 millimeters 81 00:17:12,880 --> 00:17:37,769 on those parallels so 28 and 50 in okay 15 28 now we erase the rest of the line we don't need okay 82 00:17:41,500 --> 00:18:07,660 and we are going to draw arrowheads be clean keep the accuracy and this is r 83 00:18:09,420 --> 00:18:27,230 sorry that's r15 and that's r28 please keep the position of the numbers okay 84 00:18:27,230 --> 00:18:42,190 these are vertical and they always turn to the left okay good and now we are going to draw the 85 00:18:42,190 --> 00:18:52,890 parallels for drawing the parallels you need to place the first ruler on s second ruler 86 00:18:52,890 --> 00:19:07,559 you place is beside the other one okay so beside the other one like this okay something like that 87 00:19:07,559 --> 00:19:15,319 and now i'm going to zoom in again the second one keeps still this one keeps still 88 00:19:15,319 --> 00:19:30,289 and you move it to the measure 15 like that parallel line and you do the same on measure 89 00:19:30,289 --> 00:19:39,630 28 both parallel lines you need the parallel line to cross the d-angle bisector okay that way 90 00:19:39,630 --> 00:19:56,059 where the parallel crosses the angle of a sector you get the solutions o1 and o2 but this is not 91 00:19:56,059 --> 00:20:04,660 solved yet because we need the tangency points to exactly solve this the problem okay for the 92 00:20:04,660 --> 00:20:15,799 tangency points to place you need to do this you are going to draw from perpendicular to line r 93 00:20:15,799 --> 00:20:24,859 passing through o1 right angle okay look at the ruler so right angle 94 00:20:24,859 --> 00:20:33,900 tangency point and you do the same perpendicular to r passing through 95 00:20:33,900 --> 00:20:48,680 0.02 so you have two perpendiculars here and here and you do the same with line s with the 96 00:20:48,680 --> 00:20:57,279 horizontal be careful and don't do this okay this is wrong you need a perpendicular not to the 97 00:20:57,279 --> 00:21:19,740 angle bisector but to the line so we're going to draw perpendicular to the line here and here 98 00:21:25,710 --> 00:21:37,450 okay so you have two perpendiculars there here and here so you have the tangency points 99 00:21:37,450 --> 00:21:49,640 we have the four tangency points okay good and by the way three and no one 100 00:21:49,640 --> 00:21:58,299 doesn't have anything to do okay so maybe in your in your case three is lower than than no one 101 00:21:58,299 --> 00:22:06,319 okay so because you place three wherever you want great so we are finishing now we are going to draw 102 00:22:06,319 --> 00:22:25,190 the solutions so placing the compass on 01 you have the first solution okay again so you place 103 00:22:25,190 --> 00:22:37,440 the compass on 01 and this is the radius till the tangency point and you draw the first solution 104 00:22:37,440 --> 00:22:49,359 so this is the first solution there you have a little mistake okay this should pass through 105 00:22:49,359 --> 00:22:58,750 the tangency point this is a one millimeter mistake okay and we are going to do the same 106 00:22:58,750 --> 00:23:12,609 with o2 so you draw the circumference and the mistake is here again okay why is this mistake 107 00:23:12,609 --> 00:23:19,410 happening because the angle bisector is one millimeter higher than the position it should be 108 00:23:19,410 --> 00:23:31,210 okay it's a very very normal mistake so if it happens to you don't be so worried okay 109 00:23:31,210 --> 00:23:32,849 thank you for listening