1 00:00:00,910 --> 00:00:05,960 Students in this video, let's discuss equilateral triangles in great detail. 2 00:00:06,370 --> 00:00:12,520 So far we have seen that equilateral triangles are triangles where all the sides are equal. 3 00:00:12,820 --> 00:00:16,150 So in this case, let each side be of length A. 4 00:00:17,040 --> 00:00:23,400 All right, now in an equilateral triangle, we have also seen that all the angles are equal in every 5 00:00:23,400 --> 00:00:25,350 angle is equal to 60 degrees, right. 6 00:00:25,500 --> 00:00:29,490 Because we know that the sum of the angles of a triangle are 180 degrees. 7 00:00:29,880 --> 00:00:30,360 All right. 8 00:00:30,570 --> 00:00:38,250 Now, let me draw a line from this vertex to the base such that it is splitting the base into two equal 9 00:00:38,250 --> 00:00:38,640 halves. 10 00:00:39,000 --> 00:00:48,240 Two, I draw the line over here such that BD is equal to D.C. or D, the midpoint of B.C.. 11 00:00:48,630 --> 00:00:49,130 All right. 12 00:00:49,410 --> 00:00:57,780 Now you can see that if you analyze triangle abeed, that is this triangle over here and Triangle ADC, 13 00:00:57,780 --> 00:00:59,220 that is the triangle over here. 14 00:00:59,430 --> 00:01:05,960 You can see that these two triangles are congruent Asper as as is criteria. 15 00:01:06,090 --> 00:01:06,400 Right. 16 00:01:06,420 --> 00:01:06,960 Why are they. 17 00:01:06,960 --> 00:01:11,260 So you have this site equal to this side because both is equal to it. 18 00:01:11,670 --> 00:01:20,340 We have seen that this is the midpoint of B.C. So beauty is equal to D.C. and it is a common sight to 19 00:01:20,340 --> 00:01:27,600 all the three sides of these two triangles are equal and hence, as per Essence's criteria, Triangle 20 00:01:27,600 --> 00:01:31,080 Abdeh and Triangle ADC are congruent. 21 00:01:31,460 --> 00:01:31,910 All right. 22 00:01:32,100 --> 00:01:37,470 Now, therefore, we can conclude that Angle and B will be 90 degrees. 23 00:01:37,500 --> 00:01:37,840 Right? 24 00:01:38,040 --> 00:01:38,400 Why? 25 00:01:38,610 --> 00:01:40,400 Because let me right that over here. 26 00:01:40,590 --> 00:01:50,460 Angle A B B is equal to angle E DC because they are corresponding angles of two congruent triangles. 27 00:01:50,640 --> 00:01:54,690 And you can see from the diagram that they form a linear pair. 28 00:01:54,690 --> 00:01:55,010 Right. 29 00:01:55,230 --> 00:01:59,070 And we know that a linear pair adds up to 180 degrees. 30 00:01:59,260 --> 00:02:02,550 So say this is X degrees, this is also X degrees. 31 00:02:02,700 --> 00:02:04,980 That means X plus X or two. 32 00:02:04,980 --> 00:02:09,030 X is equal to 180 degree or X is equal to 90 degrees. 33 00:02:09,180 --> 00:02:12,670 That's why this angle over here will be 90 degrees. 34 00:02:12,960 --> 00:02:13,410 All right. 35 00:02:13,590 --> 00:02:21,980 Now, if side B, C is of length E and we know D, the midpoint of B, C, therefore B is of length 36 00:02:21,990 --> 00:02:22,710 EBITA. 37 00:02:23,190 --> 00:02:29,400 Right now, over here we have a right angle triangle, right triangle beauty. 38 00:02:29,640 --> 00:02:35,360 And the hypotenuse is upside it and another side is of length EBITA. 39 00:02:35,550 --> 00:02:40,160 Therefore we can find that aid, which is the height of this triangle. 40 00:02:40,350 --> 00:02:46,100 It's equal to the square root of a square minus Ebeye to the whole square. 41 00:02:46,110 --> 00:02:46,430 Right. 42 00:02:46,680 --> 00:02:47,220 Why is this. 43 00:02:47,220 --> 00:02:49,140 So this is an upper Pythagoras theorem. 44 00:02:49,140 --> 00:02:49,410 Right. 45 00:02:49,570 --> 00:02:57,990 Because we know is square is equal to it by two the whole square plus it's square and remember it's 46 00:02:57,990 --> 00:02:58,620 over here. 47 00:02:58,620 --> 00:03:05,490 Is it right now from here I find that edge is the square root of square minus two. 48 00:03:05,490 --> 00:03:06,390 The whole square. 49 00:03:06,660 --> 00:03:09,000 All right, let me make some space over here. 50 00:03:11,690 --> 00:03:19,790 All right, now on simplifying this equation, we get that the height of the equilateral triangle can 51 00:03:19,790 --> 00:03:23,240 be described as rude prebuy to a. 52 00:03:24,530 --> 00:03:26,350 So the Heidi's route three by two. 53 00:03:26,720 --> 00:03:32,780 So we have found the height of the equilateral triangle in terms of the side of the equilateral triangle. 54 00:03:32,990 --> 00:03:33,430 All right. 55 00:03:33,650 --> 00:03:36,230 Now let's find the area of this equilateral triangle. 56 00:03:36,470 --> 00:03:40,130 We know that area of a triangle is half base in the height. 57 00:03:40,130 --> 00:03:40,490 Right. 58 00:03:40,850 --> 00:03:41,260 All right. 59 00:03:41,630 --> 00:03:45,380 So let me take B.C. as the base that's of length. 60 00:03:45,920 --> 00:03:48,890 And I know that the height is Route three by two. 61 00:03:49,340 --> 00:03:53,540 So that's half in to a new route three by two. 62 00:03:53,990 --> 00:03:57,080 And that gives me Route three by four square. 63 00:03:57,230 --> 00:04:02,360 And that is the equation which we can use to find the area of an equilateral triangle. 64 00:04:02,630 --> 00:04:03,770 So let's quickly revise. 65 00:04:03,990 --> 00:04:05,990 We have seen that in a little triangle. 66 00:04:06,170 --> 00:04:11,940 All the sides are equal and all the angles are equal and the angles are equal to 60 degrees. 67 00:04:12,170 --> 00:04:12,550 All right. 68 00:04:12,710 --> 00:04:20,300 And we have seen that the line segment, which is dividing the opposite side of Vertex into two equal 69 00:04:20,300 --> 00:04:23,600 halves, is also a perpendicular line segment. 70 00:04:23,750 --> 00:04:24,980 That is we have seen angle. 71 00:04:24,980 --> 00:04:32,410 It is 90 degrees later we will see that this is the median as well as the perpendicular bisecting. 72 00:04:32,460 --> 00:04:32,840 All right. 73 00:04:32,840 --> 00:04:35,720 That will be following up in future video. 74 00:04:36,020 --> 00:04:37,090 Now, let's press on. 75 00:04:37,310 --> 00:04:42,260 We have seen that the height of the equilateral triangle is equal to Route three by two. 76 00:04:42,950 --> 00:04:49,700 So in this way, we have expressed the height of the triangle in terms of the side of the triangle. 77 00:04:49,970 --> 00:04:55,580 And we have seen that the area of the triangle is equal to room three by four, a square.