Educational videos for Students in k-12 | Lumos Learning

Educational videos for Students in k-12




Transcript
00:00 this video is a basic geometry view for those who
00:03 are taking the S . A . T . Or
00:05 the H . T . Exam . And if you're
00:07 taking a geometry final exam you could benefit from this
00:10 video too . So let's go over some common shapes
00:13 and the formulas they need to know . So let's
00:16 say if we have a circle and my drawing is
00:18 not that great but Let's work with it . And
00:22 let's say that the radius of the circle is five
00:27 given the radius , what is this ? A conference
00:30 of the circle ? And also what is the area
00:34 ? The first formula you need to know , the
00:35 circumference is two pi R . So therefore it's going
00:41 to be two pi times five Which is 10 lbs
00:47 . Mhm . So that's the exact answer . Sometimes
00:50 you may need to put this in your chocolate and
00:53 get a decimal value . So you could use the
00:56 fact that pie is about 3.1416 . So this is
01:03 going to be 31 416 So that's the circumference .
01:08 Now , what is the equation for the area of
01:10 a circle ? The area of a circle is pi
01:16 r squared . So for this particular example it's pi
01:20 times five squared , Five squared is 25 , so
01:23 the area is 25 5 . So 25 times I'm
01:30 going to use the exact value As a decimal ,
01:32 that's 78 0.54 . Now what about the diameter if
01:38 you know the radius of the circle , what is
01:40 the diameter ? So the radius is between the center
01:44 of the circle and it touches any point on the
01:47 circle . That's the radius . The diameter also passes
01:52 through the center of the circle but it's a distance
01:54 from one edge of the circle all the way to
01:56 the other edge and it always has to pass through
01:59 the center of the circle . If you have a
02:02 line that touches two points in a circle but doesn't
02:05 pass through the circle , I mean the center of
02:07 the circle that is , it's a cord . So
02:10 this line it touches these two points on a circle
02:13 but it doesn't pass the center of the circle ,
02:15 so that makes the court . But a diameter is
02:19 between two points on the edge of the circle and
02:21 the diameter passes through the center of the circle .
02:24 So make sure you know the difference between the diameter
02:26 and the court . The diameter is twice the value
02:29 of the radius is to our , So in this
02:32 case two times 5 is 10 . So for a
02:35 circle these are the three main equations , you need
02:37 to know the circumference two pi r the area pi
02:41 r squared and the diameter is twice the length of
02:44 the radius . Now the next shape that we're going
02:48 to talk about is the square . So let's say
02:52 that The side length of the square is eight .
02:57 What is the area ? And what is the perimeter
02:59 of the square ? The area of a square is
03:04 basically side squared . All sides of the square are
03:09 the same . So in this case asks is eight
03:12 . So the area is just gonna be eight squared
03:14 , which is 64 square units , not to find
03:18 the perimeter . The perimeter is the sum of the
03:21 four sides , it's S plus S plus S plus
03:25 S . If you add S four times , it's
03:28 the same as four times s So the perimeters four
03:31 times 8 . So it's 32 units long . So
03:35 make sure you know those two equations for square ,
03:37 the area is side squared . The perimeter is simply
03:40 the sum of all four sides . Or for us
03:45 . Now , here's a question for you . Going
03:48 back to the square , let's say the area is
03:51 36 square feet , What is the perimeter of the
03:56 square ? So we know that the area is s
04:02 squared by the way for each of these questions ,
04:04 pause the video and see if you can figure it
04:07 out . So the area is 36 . If we
04:10 take the square root of both sides we can get
04:14 the life of each side So each side is six
04:18 units long . Therefore the perimeter is six plus six
04:23 plus six plus 64 times Or four times six .
04:26 So it's 24 ft long . That's the perimeter .
04:32 Now going back to the circle , let's say if
04:39 we're given the circumference of the circle , Let's say
04:42 the circumference is 16 lbs . With this information ,
04:46 find the left of the diameter and also the area
04:49 of the circle . So first we need to find
04:53 the radius . We know that the circumference is two
04:56 pi R . And the circumference is 165 . So
05:01 what we need to do is divide both sides by
05:04 two pi . So two pi divided by two pies
05:10 , 1 here , the pies cancel . So the
05:12 radius is 16 divided by two . So it's eight
05:15 units long . If you have the radius , you
05:17 can easily find the diameter . The radius is twice
05:20 the length of the diameter , So it's 16 units
05:24 . So now we can find the area which is
05:25 simply pi R . Squared . So it's pi times
05:29 eight squared Or simply 64 pi . So that's the
05:34 area . Now let's say if you have a rectangle
05:38 And let's see the left of the rectangles , 10
05:41 And the width is five . What is the area
05:44 and what is the perimeter of directing ? Feel free
05:48 to pause the video and find these two things .
05:52 So this is the left , This is the west
05:55 , this side is also the witnesses . The length
05:57 , the area is simply lap times with the perimeter
06:00 is two . L plus two . W . So
06:03 the area is going to be 10 times five .
06:06 So it's 50 square units . The Perimeter is going
06:10 to be two times 10 Plus two times 5 .
06:14 Two times 10 is 22 times five is 10 ,
06:17 20 plus 10 is 30 . So the perimeter is
06:22 30 units long And the area is 50 square units
06:30 . So let's say if the area is 40 units
06:39 long and let's say the length is eight units .
06:47 What is the perimeter of the rectangle ? Go ahead
06:51 and try that problem . So the left is eight
06:55 . We don't know the with but we could find
06:58 the winner by using this equation . So 40 is
07:01 equal to eight times w . So w . is
07:04 40 divided by eight , so it's five minutes long
07:07 . And then once you have the with you can
07:12 now find a perimeter , this is going to be
07:15 two L plus two W . So it's uh two
07:19 times eight plus two times five And at 16 plus
07:24 10 Which is 26 . So here is the practice
07:30 problem . You could try so go ahead and take
07:32 a minute and see if you can figure it out
07:34 . The left of a rectangle is three , more
07:36 than twice the whiff . If the area is 44
07:40 square cm , what is the perimeter of the rectangle
07:46 ? So take a minute and work on that problem
07:49 . So let's draw a picture . So this is
07:54 the left , this is the wife . Now let's
07:56 write an equation the length A stream or or three
07:59 plus twice the width , that's two W . And
08:04 we know the area is 44 square units . What
08:08 is the perimeter of a rectangle ? If we could
08:11 find the left and the whiff and then we could
08:14 find the perimeter , The area , we know it's
08:17 a lot of times with and what we can do
08:20 if we want to is we can replace out with
08:24 three plus two W . So we can get the
08:26 area equation in terms of W alone . So you
08:29 got to solve this by substitution . So three plus
08:32 2 W Times W is equal to 44 . Now
08:37 let's distribute W . So W time stream is www
08:41 W times to W is to w squared . Now
08:44 let's move the 44 from the left side to the
08:46 right side . So zero is equal to two W
08:50 squared plus three W minus 44 . So what we
08:54 have is the train , Omiya or quadratic expression .
08:57 And we need to factor in order to find the
08:59 value of W . So how can we factor this
09:02 particular train oatmeal ? What we need to do first
09:05 is multiplying the leading coefficient , which is to By
09:11 the constant term -44 two times negative 44 is negative
09:16 88 . So what's your numbers multiply two . Negative
09:20 88 . But add to the middle coefficient three ,
09:25 This is positive 11 and -8 . 11 plus negative
09:29 eight as up to three . But they multiply its
09:31 negative 88 . So now what we're gonna do is
09:34 we're going to replace the middle term www With 11
09:39 . w . -8 W . So it's gonna be
09:42 zero is equal to two . W squared minus eight
09:45 W . Plus 11 W -44 . I wanted to
09:51 put the 11 next to the 44 because 44 is
09:54 a multiple of 11 and eight is a multiple of
09:57 two . So , and the next step , we're
09:59 gonna factor by grouping and that's why I've arranged it
10:02 the way I did . So in the first two
10:04 terms , let's take off the G C F .
10:07 The greatest common factor is to W two W squared
10:12 divided by two W . That's going to be W
10:15 -8 W divided by two . W is negative four
10:19 In the last two terms . Let's take out an
10:22 11 and let's get rid of some of this stuff
10:25 over here . Actually , I'm gonna need that solicitous
10:32 , get rid of this and I don't think I
10:35 need this for now . So we take out an
10:41 11 , 11 w divided by 11 is W And
10:46 -44 divided by 11 . That's negative for So now
10:51 let's factor W -4 . When those two terms are
10:55 the same , that means that you're on the right
10:57 track , you've done everything correctly so far . So
11:01 if we take out W -4 from this term ,
11:04 What we're going to have left over is the two
11:05 W . And if we take it out from the
11:07 second term We're going to have 11 left over ,
11:10 but it's gonna be plus 11 . So now what
11:13 we need to do is set both factors W -4
11:16 And so w plus 11 equal to zero . So
11:20 if we add four to both sides we can see
11:22 that W is equal to four . And the other
11:26 equation we gotta start by subtracting both sides by 11
11:30 , so W . Two W . Is equal to
11:31 negative 11 . And if we divide by two W
11:35 is -11 over to now we're going to get rid
11:38 of the negative answer because we're dealing with a real
11:40 life object And to have a side length of negative
11:44 5.5 doesn't make sense . So we're going to choose
11:47 this value W . Is equal to four . So
12:02 if W . Is for We can now find the
12:05 length which is 3-plus 2 W . Or three plus
12:09 two times four . So that's three plus eight and
12:12 that's 11 . Mhm . So the left is 11
12:15 and the wife is for And we can see why
12:18 the area is left times with four times 11 Which
12:21 is 44 . So that works out now . We
12:24 can find the perimeter , the perimeter is two L
12:26 . Plus two W . So that's two times 11
12:31 plus two times for it . And so that's 22-plus
12:35 8 which is 30 . So that's the answer to
12:39 this particular problem . That's the permanent of the rectangle
12:42 by the wing . For those of you who are
12:44 taking the A . C . T . Exam or
12:46 the S . A . T . Exam . When
12:49 you get a chance , go to Youtube and search
12:52 out my ACT math video and satellite map video .
12:55 You can get more examples than multiple choice practice problems
12:58 . If you want to practice and prepare for the
13:01 mass sections of those exams , I'm also going to
13:04 post it At the last 20 seconds in the end
13:07 of this video . So you can find the link
13:09 there as well or you just search it to YouTube
13:11 . You should come up but let's continue with this
13:14 problem . The left of a rectangle is three more
13:18 than its with . If the Perimeter is 26 what
13:22 is the area of the rectangle ? So go ahead
13:25 and try that problem . Now we know the perimeter
13:29 is two L . Plus two W . So 26
13:35 is equal to two hour plus two . W .
13:38 Now notice that we could simplify this equation . Let's
13:42 divide everything by two . So 13 is equal to
13:48 L plus W . Our goal is to find the
13:51 area of the rectangle . If we could find the
13:54 dimensions , if we could find the left and the
13:56 wife then we could easily find the area . Now
13:59 we're told that the length is three more than its
14:02 with . So L . Is three plus W .
14:06 So we're going to do at this point is replaced
14:08 out With three plus W . So 13 is equal
14:12 to three plus W plus W . So that's three
14:17 plus 2 . W . Now we can find the
14:19 value of W less , attract both sides by three
14:23 , so 10 is equal to two W . And
14:26 if we divide both sides by two , 10 divided
14:29 by two is 5 . so w . s .
14:31 five L . A . Stream more than W .
14:34 So three Plus 5 is eight . So L .
14:39 S . A . So we have the with and
14:43 we have the left . Now we know that the
14:46 area is lifetimes with Or eight times 5 . So
14:51 it's 40 square units . And that's the answer .
14:57 Now let's talk about triangles . Let's say if we
15:00 have a right triangle and was saying One side ,
15:05 one of the legs mystery and the other leg is
15:07 four . What is the high partners of the triangle
15:12 ? Yeah . Let's call this side A . B
15:16 . And C . According to the protagonist . Um
15:19 A squared plus B squared is equal to C squared
15:23 . So we can say A stream B . Is
15:25 for . And let's find see three squared is 94
15:29 squared is 16 And nine plus 16 is 25 .
15:34 So if you take the square root of 25 ,
15:38 This will give you five . So the length of
15:40 the hypotenuse which is the side across the 90° angle
15:44 represented by this box . That is five minutes long
15:50 . Let's try another example . Let's see . The
15:53 hypotenuse is 13 minutes long And one of the legs
16:02 is five . Find miss inside . And I need
16:06 to know some special numbers for a right triangle .
16:10 There's the 345 triangle , There's the 5 12 13
16:15 Triangle , Which is the one that we need to
16:18 notice that the missing side is 12 . And some
16:20 other ones they need to know is the 7:24 25
16:24 triangle . The 8 1517 Triangle . There's the 9
16:30 40 41 triangle And also the 1160 61 Triangle .
16:35 So if you know these triangles , you don't have
16:37 to use the pythagorean theorem formula . You can just
16:40 simply find the missing number . But let's confirm it
16:44 . Using Pythagoras there , Let's prove that this is
16:47 12 . So we know that a squared plus B
16:54 squared is equal to c squared . And our goal
16:56 , let's say we're looking for B . So A
16:59 . Is five . And the high pattern you see
17:01 is 13 five squared is 25 13 square 13 times
17:07 13 is 1 69 . And if we subtract both
17:10 sides by 25 , 169 -25 is 1 44 .
17:16 So now we've got to take the square root of
17:17 144 which is 12 . So it pays to note
17:22 the special by triangles . Here's another example . So
17:30 let's say that the hypothesis is 10 And one of
17:36 the sizes six . What is the miss inside ?
17:40 Go ahead and pause the video and figure it out
17:44 soon . Let's write the special triangles 3455 12 ,
17:50 13 , 8 , 15 , 17 and so forth
17:54 . Notice that if we take the 345 triangle and
17:57 if we multiply everything by two , Notice what happens
18:03 , we're going to get the 6810 triangles . So
18:07 multiples of the 345 triangle also apply to a right
18:11 triangle here . This is six . This is 10
18:14 . The missing number is eight , so X .
18:16 Is eight . So let's prove that . So a
18:19 squared plus B squared is equal to see square A
18:23 is six . We're looking for being . And the
18:25 hypothesis 10 six squared is 36 . 10 squared is
18:30 100 100 -36 is 64 . And the square root
18:34 of 64 will give us the missing side which is
18:38 eight . Yeah . So let me give you a
18:42 few examples for all of these . Find the miss
18:45 inside , find the value of X . You can
18:52 get started with the first example . Okay , so
19:43 let's start with the first one . The first one
19:46 is associated with the 7 24 25 triangle . Therefore
19:50 X . is 25 . Now for the 2nd 1
19:54 It's associated with the 8 15 17 Triangle , We
19:58 have 15 and 17 . So the missing side has
20:01 to be eight . Now for the 3rd 1 notice
20:07 that It's proportional to the 345 triangle , It's not
20:12 proportional to any other right triangle . So if we
20:15 multiply these numbers by three , we're going to get
20:18 two of the numbers that we have in the triangle
20:20 three times 3 is nine , Four times three is
20:23 125 times streets 15 . So if you see two
20:27 sides present , then you know , the third side
20:30 has to be 12 . The next one is simply
20:34 the nine . It's the 40 . Actually , I
20:38 messed up on this one . I meant this to
20:40 be nine . They're supposed to be the 9 40
20:42 41 but I got mixed up with the 11 60
20:45 61 soon . That's a bad problem , ignore it
20:49 . Now , the next one is associated with the
20:51 5:12 13 triangle . If you multiply those numbers by
20:56 two , It's gonna be five times 2 is 10
21:01 , 12 times two is 24 . 13 times two
21:05 is 26 . So we have two of these numbers
21:09 10 and 26 . So the missing side has to
21:11 be 24 . Now for the last one maybe dry
21:16 because of all the clutter . This is 3040 .
21:21 And we're looking for X . Notice that this is
21:23 similar To the 345 triangle . So if you multiply
21:28 three by 10 we get 34 by 10 40 .
21:31 So five times 10 is 50 . So that partners
21:34 is 50 . Here's another problem that you could try
21:37 . A rectangle . A B C D . A
21:41 B is 12 units along And AC . is 13
21:46 minutes long . What is the area of the rectangle
21:49 ? Now rectangles , they form a 90° angles and
21:53 look at the triangle that forms we have a right
21:55 triangle If this is 12 and that's 13 . What
21:58 is the most inside ? So we know this is
22:00 the 5 12 13 triangle . So left B .
22:03 CS five . So now we can find the area
22:06 . The area of the rectangle is left times with
22:09 The left is 12 , the width is five ,
22:12 So it's just 12 times five which is 16 .
22:16 So as you can see you can find the answer
22:18 quickly . If you know your special by triangles ,
22:21 you don't have to use the bathroom formula . If
22:24 you commit this summer me it will save you a
22:25 lot of time . And on the S . A
22:27 . T . Test and on the T . Test
22:30 time is a factor . So you want to find
22:33 the answer quickly .
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