NMOS 2018 R1 Q26 —— Geometry Model

[Music] Hi everyone, I'm Tisha EJ. Welcome back

to our YouTube channel. Today we are going to talk about a past year question for AML 2018 which is about the geometry model. Let's have a look. So you can see

model. Let's have a look. So you can see that in the figure below each side of the square A B C D is 12 cm. So you can see that we have the square A B C D.

Then we say that the side is 12 cm. E

and F are midpoint of AB and A D respectively. So you can see that E is

respectively. So you can see that E is the midpoint of A and B and F is the midpoint of A and D. Then we say that FG is equal to 1 / 3 of FC. You can see

that FG is over here and then the whole line is FC. Then we are asked to find the area in centime square of the shaded region. So I can see that in this

region. So I can see that in this question the shaded region is this triangle A E G. All right. So based on the question we know that E and F is the

midpoint which mean that the length of all of these four part A E B A F and FD they are just 6 cm. So to find the area

of the triangle we'll need to know the base and the height. Now let's say AE is the base and we already know that it is 6 cm. How about the height then? So the

6 cm. How about the height then? So the

height is just we can draw a line from the opposite vertic toward the base. Hm.

But how do we find the height of this triangle?

It seems like we are not able to find it. Now let's try to see if we can

it. Now let's try to see if we can construct any geometric model to find a relationship. Here you can see that

relationship. Here you can see that there is a point G in the middle. Right?

The point G is in the middle of the square and it is connected to the vertices A and also C. How about we try to connect it with the other two

vertices B and also D. Now we connect point G to point B and then we connect point G to D. After we draw the line, can you observe any relationship now? So

actually there is a relationship here.

No worry. Let's look at an example.

So we start with a small rectangle. So

now what can we say about the relationship between the shaded and unshaded region?

Oh, they are the same. So what is the relationship of the shaded region and the whole rectangle?

It is half of the whole rectangle. Now

let's look at the second example. So you

see that this time we have a bigger rectangle right here. So again what is the relationship between the shaded and the whole rectangle?

Oh, it is also half of the whole rectangle. But why do we say so? So

rectangle. But why do we say so? So

actually we can try to cut our rectangle here in the middle with a horizontal line. Now you'll see that it will form

line. Now you'll see that it will form two rectangle and each of the rectangle that we have found here is just similar as the rectangle in our first example.

Now let's look at our next example which is an extra bigger rectangle right here.

Again what can we say about the area of the shaded region and the whole rectangle?

It is half of the rectangle. This time

we can cut our rectangle in the middle with a vertical line. then you will see that it will now form two rectangle.

Again the rectangle that we have here is also similar to what we have in our second example. So the area of the

second example. So the area of the shaded region and unshaded region they are just the same. So all of this example right here we actually call them the half model in rectangle. So what is

the characteristic for the model in the third example right here. So actually we say that we'll need to connect a point inside the rectangle right here to all

the four vertices and then we'll say that the area of the triangle form at the left and also the right is equal to the area of the triangle at the top and also at the bottom. Now let's go back to

our question. So we learn about the half

our question. So we learn about the half model. Can you see any relationship?

model. Can you see any relationship?

Now, oh, we can see that the triangle A B G and also B CG, they're actually half of the spring. And also the triangle A B

G and D CG, they are also half of the square. And you can see that for the

square. And you can see that for the triangle A BG there are actually two triangle right here which is B E G and the triangle that we want to find the A

E G. So we can actually say that the sum

E G. So we can actually say that the sum of area for the triangle A E G B E G and also D CG is actually half of the whole square. So now let's try to calculate

square. So now let's try to calculate what will be the area. So we'll use 1 / 2 * 12 and again multiply by 12 and

we'll get that is equal to 72 sample of the triangle A E G H. Let's try to observe our figure again. Oh, so you can see that for the triangle A E G and also

B E G they actually have the same base which is six cm.

How about the height?

They also have the same height which we can draw from the point G to the base.

So we can say that the area of the triangle A EG is the same as the area of triangle B EG. This mean that if we are able to find the area of the triangle D

CG then we just have to use the total area of this G triangle that we have just found then we just met up with the area of D CG then we divided by two we

will get the area of triangle A E G again let's try to observe our figure based on what we have from the question we know that FG is 1 / G of FC this

means that the multiple of FG and GC it is one and two and you can see that these two triangle D FG and also D CG

they actually have the same height. Oh,

so this is actually the equal height model right here. We know that the multiple of area between the two triangle is the same as the multiple of the base. So you can see that these two

the base. So you can see that these two triangle actually form a bigger triangle D CF right B CF. So let's try to

calculate it. The area of the triangle D

calculate it. The area of the triangle D FC is actually just 1 / two and then we multiply by six and then which is equal

to the 36 cm squared.

And now let's now that we have now that we have these two information that we have just calculated let's try to find our final answer for the area of

the triangle D CG based on the multiple of B. It is just equal to 36 / 3 * 2

of B. It is just equal to 36 / 3 * 2 which is equal to 24 cm squared. So the

area of the triangle a eg is just 72 minus out with the area of dcg which is 24 and then we divided by 2 which is just equal to 24 cm square and this is

our final answer right so now let's summarize what we learn in this question. So in this question we are

question. So in this question we are trying to find the area of triangle inside a square but we are not able to calculate it using the formula. So we

try to construct a geometry model to find the relationship. So the first model that we found is as shown in this figure which is what we call the half model in rectangle. So the

characteristic of this model is that we'll need to connect a point inside the rectangle to all the vertices. Then we

say that the area of the triangle formed at the left and the right is equal to the area of the triangle at the top and the bottom. The next model that we have

the bottom. The next model that we have found is as shown in the figure below which is what we call the equal height model. For triangle of the same height,

model. For triangle of the same height, the multiple of the area between the two triangle is the same as the multiple of the base. So now you can try to do this

the base. So now you can try to do this practice question and you can leave your answer at the comment section below.

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