1
00:00:00,770 --> 00:00:08,600
I students in this video, let's discuss a very important and useful property in right angle triangles

2
00:00:08,660 --> 00:00:12,030
called the Pythagoras's theorem or the Pythagoras's property.

3
00:00:12,350 --> 00:00:16,130
All right, so let's take an example of a right angle triangle.

4
00:00:16,430 --> 00:00:19,340
This angle over here is a 90 degree angle.

5
00:00:19,610 --> 00:00:28,250
And we call the side opposite to the angle that makes a 90 degree as the hypotenuse to this side over

6
00:00:28,250 --> 00:00:30,830
here is opposite to the 90 degree angle.

7
00:00:31,040 --> 00:00:33,080
And we call that the hypotenuse.

8
00:00:33,380 --> 00:00:33,870
All right.

9
00:00:34,130 --> 00:00:38,090
Now, the other two sides are legs of the right angle triangle.

10
00:00:38,600 --> 00:00:39,860
Now let's move on.

11
00:00:40,010 --> 00:00:41,500
We have defined these terms.

12
00:00:41,510 --> 00:00:43,340
Remember what the hypotenuse is?

13
00:00:43,550 --> 00:00:46,430
It's the side opposite the 90 degree angle.

14
00:00:46,880 --> 00:00:47,360
All right.

15
00:00:47,630 --> 00:00:50,180
Now let's take a right angle triangle.

16
00:00:50,390 --> 00:00:56,660
We have a triangle with sides A, B and C, and this is a right angle triangle with a 90 degree angle

17
00:00:56,660 --> 00:00:57,350
over here.

18
00:00:57,710 --> 00:00:58,160
All right.

19
00:00:58,550 --> 00:01:02,710
Let's make three more copies of this right angle triangle.

20
00:01:03,140 --> 00:01:03,770
All right.

21
00:01:04,070 --> 00:01:09,160
And let's change the arrangement of these four triangles in this manner.

22
00:01:12,280 --> 00:01:14,800
Now, let's make a copy of this over here.

23
00:01:15,550 --> 00:01:23,340
OK, now you will see that this area over here is equal to a into a that's a square, right?

24
00:01:23,590 --> 00:01:26,370
Because over here you get a square.

25
00:01:26,410 --> 00:01:41,140
Let me name this figure a, b, c, d, e, f, g, and it's now B, B if it is A square.

26
00:01:43,400 --> 00:01:46,540
And each of its sites is of lengthy.

27
00:01:46,910 --> 00:01:52,940
We know that the area of a square is a into a you can check more about it under quadrilaterals.

28
00:01:53,240 --> 00:01:56,270
Now, the area of a square is site square lower here.

29
00:01:56,420 --> 00:01:57,980
Every site is equal to eight.

30
00:01:58,160 --> 00:02:01,590
Therefore, the area of this square is equal to a square.

31
00:02:02,300 --> 00:02:02,770
All right.

32
00:02:03,050 --> 00:02:06,680
Now the site, the angles of a square are 90 degrees.

33
00:02:06,920 --> 00:02:09,600
Now, this angle will be equal to 90 degrees.

34
00:02:09,740 --> 00:02:10,700
Why would that be?

35
00:02:10,730 --> 00:02:16,640
Let me quickly show you that let's construct a line that's parallel to AC.

36
00:02:16,820 --> 00:02:23,340
And over here now, we can see that this angle will be equal to this angle.

37
00:02:23,360 --> 00:02:24,750
Let me call this angle one.

38
00:02:25,160 --> 00:02:26,990
So this is also angle one.

39
00:02:27,320 --> 00:02:31,550
And you will see that this angle over here will be equal to this angle.

40
00:02:31,580 --> 00:02:34,990
Let me call this angle to angle to why is that so?

41
00:02:35,180 --> 00:02:37,910
Because these are alternate interior angles, right?

42
00:02:38,120 --> 00:02:44,120
This line over here, which we constructed, let me call it, as Lynelle is parallel to G and it's parallel

43
00:02:44,120 --> 00:02:44,810
to AC.

44
00:02:45,020 --> 00:02:47,450
Therefore, this angle and this angle are equal.

45
00:02:47,680 --> 00:02:50,090
Similarly, this angle and this angle are equal.

46
00:02:50,180 --> 00:02:50,600
Right.

47
00:02:50,750 --> 00:02:52,940
And we started with the right angle triangle.

48
00:02:53,120 --> 00:02:53,500
Right.

49
00:02:53,660 --> 00:02:56,770
This angle will be angled to and this is 90 degrees.

50
00:02:56,780 --> 00:03:02,930
So we know that as our angles and property angle one plus angle two plus this angle over here, which

51
00:03:02,930 --> 00:03:05,210
is 90 degrees, is equal to 180 degrees.

52
00:03:05,360 --> 00:03:11,240
Therefore angle one plus angle two is equal to 180 minus 90, which is equal to 90.

53
00:03:11,450 --> 00:03:11,790
Right.

54
00:03:11,810 --> 00:03:15,860
That's why this angle is 90 degrees and each side is of length.

55
00:03:16,400 --> 00:03:19,510
That's why BATF is it square.

56
00:03:19,640 --> 00:03:24,860
And we know that the area of a square is equal to a new A or side in the side.

57
00:03:24,980 --> 00:03:27,930
And in this case, the area is equal to a square.

58
00:03:28,370 --> 00:03:28,880
All right.

59
00:03:29,090 --> 00:03:30,140
Now let's move on.

60
00:03:30,470 --> 00:03:38,960
We have made a copy of this shape over here and we have a square as the uncovered area in this case.

61
00:03:38,960 --> 00:03:39,320
Right.

62
00:03:42,440 --> 00:03:50,380
All right, now let's make some arrangement shifting in the case of these triangles and see what happens,

63
00:03:50,780 --> 00:03:53,390
I shift this triangle in this direction.

64
00:03:54,300 --> 00:03:55,610
And it goes over here.

65
00:03:57,230 --> 00:04:03,500
All right, now the boundary is the same, you cannot all right, now this triangle is shifted in this

66
00:04:03,500 --> 00:04:04,100
direction.

67
00:04:06,490 --> 00:04:09,490
And this triangle over here, I shifted down.

68
00:04:13,020 --> 00:04:20,280
All right, now we have maintained the boundary as the same right there for the area of uncovered,

69
00:04:20,640 --> 00:04:25,770
which is uncovered over here as to be equal to the uncovered area over here.

70
00:04:25,980 --> 00:04:27,790
What do I mean with uncovered area?

71
00:04:28,380 --> 00:04:31,340
It's the area that is not occupied by a triangle.

72
00:04:31,540 --> 00:04:32,090
Over here.

73
00:04:32,100 --> 00:04:34,450
You can see this is occupied by a triangle.

74
00:04:34,680 --> 00:04:36,150
This is occupied by triangle.

75
00:04:36,420 --> 00:04:37,830
This is occupied by triangle.

76
00:04:37,860 --> 00:04:39,200
This is occupied by triangle.

77
00:04:39,360 --> 00:04:43,700
The remaining area is what we call the uncovered area over here.

78
00:04:43,860 --> 00:04:48,840
We have shifted the triangle such that this is the area covered by a triangle.

79
00:04:49,170 --> 00:04:50,580
This is the uncovered area.

80
00:04:50,830 --> 00:04:51,230
All right.

81
00:04:51,390 --> 00:04:52,710
Now, what is this area?

82
00:04:52,950 --> 00:04:56,040
This is equal to being to be which is equal to be square.

83
00:04:56,190 --> 00:04:56,550
Right.

84
00:04:56,820 --> 00:05:03,480
Because you can see that this site is B and this side is B, so we are getting a square of side B over

85
00:05:03,480 --> 00:05:03,820
here.

86
00:05:04,470 --> 00:05:11,790
Similarly, you can see that over here, this site is of lengthy and this site is of Benghazi and we

87
00:05:11,790 --> 00:05:14,040
are getting a square of sight to see.

88
00:05:14,310 --> 00:05:17,460
So the area over here is equal to C Square.

89
00:05:18,510 --> 00:05:25,260
Now, because the uncovered areas have to be equal and the uncovered area over here is equal to a square

90
00:05:25,530 --> 00:05:32,370
and the uncovered area over here is equal to be squared plus C squared, therefore is square, is equal

91
00:05:32,370 --> 00:05:34,870
to B squared plus C square.

92
00:05:35,040 --> 00:05:37,380
And remember, what is a BNC?

93
00:05:37,560 --> 00:05:41,790
These are sides of the triangle with which we started, which is a right angle.

94
00:05:41,790 --> 00:05:46,860
Triangle is the hypotenuse, which is the side opposite the 90 degree angle.

95
00:05:47,100 --> 00:05:53,040
Therefore this is what we call the Pythagoras theorem that is in a right angle triangle.

96
00:05:53,190 --> 00:05:59,790
The square of the hypotenuse is equal to the sum of the squares of the other two sides.

97
00:05:59,940 --> 00:06:04,400
That is a square is equal to B squared plus C square.

98
00:06:04,590 --> 00:06:11,430
And remember, the Pythagoras property is valid only for the right angle triangles and this is the right

99
00:06:11,430 --> 00:06:12,140
angle triangle.

100
00:06:12,360 --> 00:06:13,850
That is why this is true.

101
00:06:14,040 --> 00:06:15,360
So let's quickly summarize.

102
00:06:15,540 --> 00:06:17,550
You take any right angle triangle.

103
00:06:17,550 --> 00:06:20,190
Let's take an example or let it be five.

104
00:06:20,190 --> 00:06:27,480
Let this be three and let this before this and ninety degree angle, you will see that five square is

105
00:06:27,480 --> 00:06:30,750
equal to three square plus or square.

106
00:06:31,020 --> 00:06:32,010
What is five square.

107
00:06:32,160 --> 00:06:33,150
That's twenty five.

108
00:06:33,360 --> 00:06:34,350
What is three square.

109
00:06:34,350 --> 00:06:35,040
That's nine.

110
00:06:35,220 --> 00:06:36,300
What is four square.

111
00:06:36,450 --> 00:06:41,190
That's sixteen and nine plus sixteen you will see is equal to twenty five.

112
00:06:41,350 --> 00:06:48,120
So in a right angle triangle the square of the hypotenuse is equal to the sum of the squares of the

113
00:06:48,120 --> 00:06:52,200
other two sides and that is what we call the Pythagoras property.