Geometry Introduction - Basic Overview - Review For SAT, ACT, EOC, Midterm Final Exam - Free Educational videos for Students in K-12

Geometry Introduction - Basic Overview - Review For SAT, ACT, EOC, Midterm Final Exam - By The Organic Chemistry Tutor

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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