00:00 Hello, my name is Sonday Ayase.

00:16 We're taking Mathematics, JSS, three class, we're looking at a very interesting topic today.

00:21 The say is similar shapes.

00:23 Can shape actually be similar?

00:25 Yes.

00:26 They can.

00:27 One can be bigger, one can be smaller.

00:29 They can be similar.

00:30 They could have something that makes them look similar.

00:33 So we'll be looking at some of those shapes and that's the class we'll be designed for,

00:37 that purpose.

00:38 And what's our objective?

00:40 Identifying similar shapes, tangles, rectangles, square, cube, the cube ball, identify them,

00:48 and what makes them similar to themselves.

00:51 Two, identify the presence of similar shapes in an environment.

00:55 Three, when large figures using scale factors, when large the shapes are in scale factors,

01:03 calculate the length, area, volume of similar shapes of figures.

01:10 And lastly, you could solve problems on quantitative, meaning that we're going to be looking at

01:17 everything from the reasoning, quantitative reasoning involving similar-world shapes.

01:24 The question is, when does two shapes become similar?

01:28 Two shapes are similar, only if their matching sides are in proportion, that means the sides

01:36 at which the two shapes are.

01:37 If their sides are matching together, then you can simply say the two shapes are what, similar.

01:44 And also, if their matching angles are equal, if the angles at which the both are sighted

01:51 are equal, then the two shapes can be said to be similar.

01:55 Take note of the two words.

01:57 The first one is sides, and the second one relates to what angle.

02:07 That when we take note of this two, it becomes easy to know when the shapes are similar and

02:13 when shapes are not what, similar.

02:16 So in the world, if two objects are similar to each other, one of them can be zoomed in or

02:22 zoom out.

02:23 So if they are similar, I could zoom one in and the other will be zoom out.

02:30 So, proportion, two shapes are similar in proportion, if all the dimensions are of the same

02:37 ratio.

02:38 Remember, sides, talking about proportion, if all the sides are of the same.

02:43 So for example, if the length and breadth of B are three times the length and breadth of

02:47 A. So A and B are in proportion, meaning A and B are in proportion because A, A, the length

02:55 of B is three times that of A. We're still going to be looking at that.

03:00 But in the main sense, that's proportionality, or two of them are similar.

03:04 In terms of the ratio, a ratio states what the relationship B between two quantities.

03:08 That's for example, the ratio of A, length to B, is ratio one, ratio three.

03:14 So we can also say that the two is the two are similar.

03:17 Now look at an example of this.

03:21 We have the cubes are similar to each other, true or false, that's the first question.

03:27 The cubes are similar.

03:28 We have three cubes there.

03:30 We have a bigger one, we have a smaller one, we have another smaller one.

03:35 We would zoom in the bigger one, three, three are they similar?

03:41 Yes.

03:42 How do you know?

03:43 Check the number of sides.

03:44 The first one has many sides, one, two, three, one, two, three.

03:49 The second has one, two, three, one, two, three.

03:52 So the sides in the three are the same.

03:56 So because they are the same, the cubes are similar to each other.

04:01 Inrespective of how it has been zoomed out.

04:05 Now, the next question is the plastic balls are not similar to each other.

04:11 The plastic balls are not similar to each other.

04:14 Look at the plastic balls, look at the balls, look at the second, look at the third, look

04:20 at the fourth, look at the fifth.

04:23 The plastic balls are not similar to each other, is that true?

04:27 That's not true.

04:28 That's false.

04:29 If you look at them, the difference between these plastic balls is just the color.

04:33 But the second fence of the plastic balls makes them somewhat similar.

04:40 Looking at the diagonal, they say the diagonal below shows two triangles.

04:43 Now look at the two triangles carefully.

04:46 What do you observe?

04:47 You observe one is bigger, one is smaller.

04:51 Looking at the base, the base of the first is 12.

04:55 The base of the second is 36.

04:58 It means that the base of the second, which is B, is 3 times that of A.

05:06 The base of the B is 3 times that of A.

05:10 So if you multiply this, that's 3 times that of A, that's 12.

05:18 That should give me 36.

05:21 So the base is 3 times that of A.

05:24 Now looking at the height, the height also is 3, the height of the bigger one, which is

05:31 B, is 3 times that of A.

05:35 So 3 times the height, that's 5, that will give me 15.

05:41 Please take note.

05:42 It's very, very important.

05:43 Once you can be able to identify their heights and see what is different between them.

05:49 You could down the time and now, is these two triangles similar?

06:01 Looking at the diagram again of the keyboard, now look at the two.

06:06 At the similar, yes they are, look at the length, the length, the breadth, and look at it carefully.

06:14 You will see that the smaller one is 8 cm, this one is 60, the bigger one is 16 cm, so

06:21 twice.

06:22 That means the smaller one is, the smaller one is twice the big.

06:25 I have the bigger one, I have the smaller one.

06:28 So two of the smaller one makes the big one.

06:31 So if you have 8, this is 4, 4 times 2, that's 8, then we have 12.

06:38 That's 6 times 2, that's 12, so they are similar.

06:42 Now, looking at the diagram of a 3D, 3D cube, is this similar?

06:47 First, before you answer that question, look at the sides.

06:50 At the sides the same, at the sides, I have 5 cm, I have 25.

06:55 It means that this is 4 times bigger than the smaller one.

07:01 So I have 5 times 4, that will give me 25.

07:04 I have 4 times, I have 5 times 5, that will give me 25.

07:08 I have 5, 4 times 5, that will give me 20.

07:12 Now looking at these 3D diagrams, are they the same?

07:16 No.

07:19 Technute, very, very important.

07:21 This gives me 25, 25, 25, but in the smaller one I have, 5, 4 and 5, they are not the same,

07:30 because one of the height there is not reproducing itself.

07:35 Technute, also very important.

07:37 Now, what is enlargement?

07:39 When you talk about you enlarge, enlarge simply, it's an enlargement of a diagram, shapes

07:47 or object is a copy of the original, in which everything is made larger or keeping the

07:54 same proportion.

07:56 You pick an object, ask you an example.

08:04 I have this is the object, I decide to enlarge it to this.

08:13 I have another enlarged one to this.

08:17 Now, what we observe, enlargement, I could have A, B and C, enlargement, picking it and

08:30 making it a bit bigger, but keeping the proportion in size, I've been able to keep the proportion.

08:37 Because apparently I have 3 rectangle given, yeah, rectangle 1, 2 and 3.

08:45 So if for example, I'm to find the area of this, I'll find the area of the bigger one,

08:51 area of the nest and area of the tot, that is what enlargement.

08:56 Now, the next one is what the scalar factor, very important.

09:00 The scalar factor is the amount by which every dimension of the given shape is multiplied

09:07 to get the dimension of the enlarge, represented by K. That means, by what percentage can you,

09:14 no, this is the length, this is the breadth.

09:18 For example, by what percentage can you multiply this, to get the length, this is L1, this

09:24 is L2, this is L3, L2.

09:28 Now, by what percentage can you multiply this to get L3?

09:31 Once you are doing that, you are doing what we call the scalar factor.

09:34 This is B1, for example, this is B2, this is B3.

09:39 By what percentage or by what dimension can you multiply B1 to get B3?

09:47 That is called the scalar factor.

09:50 So, okay, by what percentage? Once you are multiplying and we represent it with that with

09:54 what? With K. Now, go.

09:58 We'll move into, let's now solve a question.

10:04 Find the scalar factor, that's K. Find the scalar factor, K, of the triangles given.

10:10 If you look carefully, I have these triangles given, every size much is size.

10:17 Every side, much the size is related.

10:21 So, if I'm to find the scalar, check the two shapes and if they are find anything that is

10:27 making them similar, meaning, find each of them.

10:31 That's by dividing the longest side of the new, by of the triangle, by the longest side

10:38 of the small, I will simply say, if you look at this, there are three, there are two sides,

10:43 I have the longest side.

10:44 So, if I'm going to be looking for, I will just say, longest, I'm looking for the longest,

10:51 first one is longest, longest sides, I'm looking for the longest sides.

10:57 So, what would I simply do, yeah? Longest sides, that's my first.

11:03 It would be, longest sides of first, that the longest sides of the first one, longest

11:12 sides of the first, that is bigger, over the longest sides of the smaller one, that is

11:23 one that is smaller of the smaller.

11:28 So, now, what the longest side of the big one, the longest side of the big one is what?

11:37 That gives me 22.5 over, sorry, longest side, that will give me 20, longest side of the bigger

11:48 one, that will give me 22.5, that's mm, mm, mm, that's 22.5 over the longest side of the

11:57 smaller one, what longest side of the smaller one, that's 9.

12:01 Because of this point, decimal place, you multiply by 10.

12:05 So, when you multiply, because of the decimal place, to take it off, you are going to be having

12:09 2.5, 2.5, 2.5, remember, I said, that's where we must go one, we must go one of this, we must

12:22 go two, we must go two, is divide the shortest side, we must go two, is divided in the shortest

12:32 side now, shortest side, shortest sides, picking the first shortest sides, shortest sides,

12:43 that goes second, the shortest side. So, you see, the shortest side of the bigger, shortest

12:56 sides, shortest sides of the small, the smaller one, of the small, now, now, if you look at it

13:12 very well, you could also go back to check off what you have, shortest sides, shortest

13:20 sides, what is shortest sides of the big, look at it, shortest side of the big is given to

13:25 be what, to be given, 12.5, and what the shortest sides of the small, shortest side of the small

13:32 is given to be what, 0.5 divided by 5, like I said, you could multiply by 10 to take away

13:39 this point, so that will give us automatically another 2.5, now, take note, the sides of the

13:51 first, give me 2.5, the sides of the second, give me 2.5, now, take note, very important,

14:03 there will divide the torched sides, that is the torched side of the triangle, you divide

14:09 the torched sides, and what the torched side, now, you will divide the longest, that is, this

14:15 longest divided by longest, we will divide this shortest by this shortest, now, we take the

14:22 other side by this other side, so you could have a complete factor, factored out, so let's

14:31 add the complete, so we have the other side, the torched side, that's from Africa 3, those

14:36 add the torched sides, and what the torched sides, the torched side for the longest one is

14:40 giving to be what is already 20 over 8, and that gives me what 2.5 again, 2.5, so the question

14:52 is 2.5, now, this means that the shape angle P, M, N is 2.5 bigger than the given shape

15:07 triangle J, K, L, take note, by what the K factor, the K factor means it is 2.5, means that

15:18 this is 2.5 bigger than the other sides, that's the word, the scalar factor, after find your

15:27 scalar factor, now, let's have an enlargement, one you enlarge, what does it mean, an enlargement

15:36 like I said before, is when the scalar factor is a fraction between 0 and 1, lead to a new

15:42 shape or object, that is smaller than the given shape or object, that is one side of the new

15:49 side is K multiplied by the matching side of a given shape, what does that simply mean,

15:57 it means that there are sometimes your shapes may not be given numbers, and they will actually

16:02 to find the sides that are missing, looking at this 2 side, rectangle STUV, it's an enlargement

16:09 of rectangle ABC, look at it, ABCD rectangle, an enlargement of it gives us STUV, an enlargement

16:23 of it, so the question is, calculate the length and the breadth of the rectangle of the

16:29 largest size, if the K is already given, the K is already given to be 3, that means your

16:36 scalar factor is given to be 3, now, actually to find the length and the breadth of this,

16:42 the first of all you write down the rule, the rule is the new length multiplied by K, that

16:49 is the new length you want to get multiplied, the new length, the new length of the shape

16:54 is equal to K multiplied by the length of the given shape, the new length you want to

17:01 get, is equal to K multiplied by the length of the given shape, that means what does that

17:08 mean, substituting what does that mean, what is the length that was given before, so we just

17:16 simply say I'm having an up and I'm having new length, so if I say new length, new L,

17:27 put it will give me my scalar factor K multiplied by old, put it this way, old L, so the new L,

17:38 new length will be my scalar factor is 3, multiplied by what is the length given before, I'm

17:44 having old length as AB, so that will just give me 3 multiplied by what's AB, that's 25,

17:52 so going fully well, looking at it, substitute a scalar factor is 3 and the length, so 3,

18:00 I'm having the length, that is the breadth, that's 3 times, 3 times, 3 times 25, that gives

18:11 me 75, 75, M, M, that's millimeter, also if I want to do the same thing to the breadth,

18:27 I would just simply say my new breadth that's B equal to my scalar K multiplied by old, old

18:37 breadth B, you look at this thing, are we giving the old breadth yes, old breadth BC is the old

18:46 breadth, so what would I just simply have, I would just simply say my 3 multiplied by my 12,

18:55 so that's what that's the, that's what we have, so substituting it, we're going to have

19:01 the new breadth, that will be 3 times 12 and that gives us what, 36, millimeter, now if the

19:10 scalar is 3, the length of the rectangle STUV is 75, M, M and the breadth is 36, 75 and 36,

19:25 now let's now go to another important aspect, that means using scale factor to calculate

19:32 areas and volume, using the scale factors, which is the K, this K, to calculate for your

19:41 area and volume, how do you do that, now two similar right angle triangles are given,

19:47 now we have two right angle triangles, we have triangle D, E, F, look at, D, E, F, it's an

19:56 enlargement of triangle B, A, C, triangle D, E, F is enlarged, out of triangle A, B, A, C,

20:10 first thing is techniques of the question, when you said it's been enlarged, you just know

20:14 that the scalar factor has to be there, they have to tell us how it has been enlarged, which

20:18 is the K, now and the scalar factor is 2, good, so the scalar factor is given, if the scalar

20:25 factor wasn't given, you start brainstorming your head, how do I know how many, what is

20:29 the dimension to which it was enlarged, so I know it was enlarged by 2, good, now we have

20:35 to go to calculate the area of the two triangles and compare your answers, calculate the area

20:42 of the first, calculate the area of the second, compare your answers, now it simply means

20:49 that you use the value of K in your comparison, calculate the area of the triangle, now first,

20:57 in this category it means you have to fail over calculate the area of the first triangle,

21:03 the first triangle is triangle, triangle, B, A, C, now the formula is 1 over 2 multiplied

21:13 by base, multiplied by height is the area of each triangle, now that will give us 1 over

21:24 2 multiplied by what's the base, the base is 16 multiplied by what's the height, the

21:33 height is 12, that's for the triangle A, B and C, you also do the same thing for the triangle,

21:41 the second triangle, and the triangle D, E and F, apply the same rule, triangle D, E and

21:53 D, E and F, now it just means 1 over 2 multiplied by the base, what is the base, the base is 32, multiplied

22:03 by what is the height, the height is 24, now when you look at this, what you are just simply

22:09 doing is treating them separately, then make comparison, so the first one gives me 96, 96

22:21 centimeter square, areas is in square, volume is in cube, so take note, now we have 384 centimeter,

22:37 okay, let's put it down, put it down, making it more easier, take it up, we have equal to

22:46 384 centimeter square, the first, the second, now if we look at this carefully, we will

22:59 arrive at the area of the first, area of the second, now how do we compare, remember we

23:09 said if you are comparing, the bigger one is at the top, the smaller one is down, when we're

23:15 doing our cake, we say the smaller, the length of the bigger size, up, the depth of the

23:20 smaller, down, so from here, simply how do you compare, how do you compare, so you will

23:27 say area of the bigger one, area of the triangle, DEF, over the area of the smaller one, at the

23:43 triangle, BAC, that's remember from the scalar, from the scalar that we meant comparison, now you

23:54 will see why the scalar of 2 can also be checked, so that gives me 384, the bigger one, over 96, now

24:07 if you look at 96, so answer gives me water, 4, it means that with this 4, I'm going to be writing

24:22 the smaller one, remember the smaller area is 96, it means that it is multiplied by 4 times the

24:32 smaller one, square, the bigger one is multiplied 4 times the smaller one, but in this proportion, this

24:39 4 can say 2 is to power 2, multiplied by water by 96 centimeter square, remember what's

24:49 okay, as scalar factor is given to be water, 2, so by what percentage is this enlarged, by water,

24:56 2, again, so the scalar factor is a checkmate to your financial answer, to know whether you are on

25:05 the same track or you have 2x times the area of the given triangle, so it is twice bigger to the

25:15 area of the small hour, given that the k factor is already water, so we checkmate your answers with

25:22 these 2, if it cannot be in this form then your answer is wrong, now the last part is applying it

25:31 to water problem, how do we apply it to water problem, how using your scalar factor to solve a

25:38 problem, olatical wants to find out the height of a lamp pool, olatical is 1.5 meter tall, this is the

25:53 diagram, it's 1.5 meter tall, it wants to find out the length of the lamp post, it has another

26:04 lamp post, it wants to find out the length, how long is the length from him, so it wants to find out the

26:10 length of it, and it's 1.5 meter tall, the length of olatical shadow changes throughout the day,

26:18 its length changes throughout the day, now olatical stands in the shadow of the lamp pool, stands in the

26:25 shadow of the lamp pool, shadow of the lamp pool, out moving, shadow of the lamp pool, so that it's

26:33 shadow and the lamp pool shadow are in a straight line, you could see, look at this shadow, meeting with

26:40 the shadow of the lamp pool at a straight line, between yeah, meeting together, now when they do that on a

26:47 straight line, the shadow of the lamp pool, and that of olatical forms a triangle, take a look at what,

26:56 forms a triangle, and is shown in the diagram, this is the diagram, olatical is standing, it wants to know

27:05 how tall it is, the lamp pool is there, from the side to olatical is 2 meter, then the entire length from

27:14 from where it was to where the pole is is 6 meter. Now we have 3 and 2 triangles here formed.

27:24 Ola de Poe's shadow is 2 meter long and the 2 meter long and the shadow of the lamp is 6 meter. Back, the shadow of the lamp, the shadow of the lamp is 6 meter. Ola de Poe's shadow is 2 meter.

27:41 It's represented draw a diagram showing the triangle formed by Ola de Poe and the shadow and a separate diagram showing the lamp poe and the shadow. Use your diagram and find out the scalar factor of the 2 triangles to help Ola de Poe walk out the eyes of the pool.

28:03 2 triangles are formed, took, put them in separation. The triangles for the lamp poe and the triangles for Ola de Poe. Now when you look at these 2 triangles very well, you have the shadows, shadows of Ola de Poe is 2 meter. The shadow of that of the lamp poe is 6 meter.

28:27 The eye of Ola de Poe is given to be 1.5. So each spot, the eye of the pole is not given. So that gives you a clue to what you are expected to find out.

28:38 Now find the scalar factor. The scalar factor is CA over CF. What does that mean? It means remember from the scalar factor.

28:50 We said that solution. A scalar factor is that of the longest. The longest, the first side given, the first side given is the wall.

29:03 The longest, remember, over longest of the beak, those sides that are given, and the longest of the longest of the bigger sides and the longest of the smaller sides.

29:32 Now the longest sides are not given. So just say the side bigger side was the most right. So what is the bigger side? That will give you a 6 divided by all the other sides that you have, that is given, you have 2.

29:44 That will give you a divided by that will 3. So your K factor, your scalar factor is 3. That means the shadow of the lamp, compared to the shadow of Ola de Poe is 3 times.

30:01 Take note of that. Now calculate the height of the lamp. Now we want to find the height because what is shadows now, we will be able to get the shadows.

30:10 Now calculate the height of the lamp. The lamp is represented by B.A. Look at the lamp. We have the height of Ola de Poe, but we don't have the length of the lamp. The lamp is represented by B.A. in the diagram. We have the height of Ola de Poe, which is what? E.F. in the diagram.

30:25 We have the height of the lamp. We have the height of the lamp. We have the height of the lamp. Remember, when we are doing that, that is your B.A., the new one you are looking for, your scalar factor multiplied by the old one you have, the old one that you have.

30:51 So what do we have? K multiplied by... What is the old one that you have? You have 1.5. It is the old one. The old one is given to be 1.5. 1.5.

31:04 So remember, we have gotten our K. What is our K? Our K has already been arrived at that. What is our K? 3. That is 3 multiplied by 1.5.

31:15 Now we multiply 3 by 1.5. What do we have? We have 4.5 meter. The height of the lamp is 4.5 meter.

31:34 The height of the lamp is 4.5 meter. The height of Ola de Poe is 1.5. What does that tells you? It means that the height of the lamp is what? 3 times that of Ola de Poe's height.

31:49 And why the shadow of the lamp is 3 times that of what? Ola de Poe's shadow. With that, we will be able to solve the question.

32:00 Now we have the evolution. Object J and K, shown below are cubes. They are two cubes. J and K. And QK is an enlargement of QJ. But where K is given to be 4? Sign the enlargement.

32:22 Calculate the length of one side of the cube, K. Calculate the enlargement of one side of the cube K. How do you do that? You look at the size of the small amount that is given compared to the size.

32:36 Calculate the area of one phase of cube K, area that is the area of the cube. Then calculate the volume of cube K. You calculate the volume of cube K.

32:49 Thank you very much, I come to the end of the class. My name is Oyase Sondi.