Transformasi Geometri Bagian 2 - Refleksi (Pencerminan) Matematika Wajib Kelas 11

Assalamualaikum warahmatullahi wabarakatuh meet again with me Denny Handayani on the Madlab channel in this video we will learn the second geometry transformation, namely reflection or reflection before we have discussed the shift or translation for friends who have not seen the previous video please check the link in the video description

In this video we will learn the second geometry transformation, which is reflection or reflection I will not give the formula directly, but we will look for it ourselves So if you forget, you can look for it yourself later

We will take an example, friends, pay attention to the following Cartesian diagram, we give the number, we take a point, the position is free, yes, I just take the position here, for example, this is the point, the coordinate is 4.2, first we will mirror this point first, this point is the source of X, so imagine that X is the mirror and remember the mirror concept

If we mirror an object, then the distance of the object to the mirror must be equal to the distance of the mirror to the image. Here we are talking about a flat mirror. So if the A here and the X axis are mirroring, the distance is two units. Then the image must be here. This is the image. The A axis is here.

Now, pay attention to the coordinates, at first a is 4.2, after we reflect to the source x, the image is 4,-2. Here the absciss or the value of x does not change, it's the same 4, the change is the value of y from 2 to -2.

So the conclusion is generally if we reflect a point x, y we reflect it towards the x-axis then the image is x, negative y so if it is reflected towards the x-axis only the value of y changes the sign so for example another example friends have a point 2.6 we reflect towards

x-axis, then the shadow is how much? the coordinate is 2,-6 yes, the value of y is the only coordinate that changes the sign okay, that's the reflection of the x-axis now we reflect it towards the y-axis, consider this y is the mirror, the y-axis is the mirror, how far is this? 1, 2, 3, 4 right? to the mirror, then the shadow is definitely here, it's 4 more, here

This is a point A image if we mirror it to the source Y. Look at the coordinates, at first it was 4.2, it became negative 4.2. What's the conclusion? It turns out that the value of X that changed, which was originally 4, became negative 4. The value of Y remains the same.

So in general, point A, X, Y, we reflect towards the sum Y, then the imagination is negative X, Y. Remember, towards the sum X, the value of Y changes. Towards the sum Y, the value of X changes. Example, 6, negative 4, if we reflect it.

towards the source Y, then the one that changes the X becomes negative 6, the Y remains negative 4. Simple, right? Okay, that's towards the source Y. Now, if we mirror it towards the line X = Y, X = Y is here. This is the line X = Y. Imagine that we mirror this point A towards this line, then where is the image? Here.

This is the shadow. So the distance of the shadow, the distance of the object to the mirror, must be equal to the distance of the mirror to the shadow. And this is straight. So, the initial value of 4.2 after being mirrored to Y = X becomes 2.4. How does it change? 4.2 becomes 2.4. Notice that the value of X and Y are exchanged. So in general, point A , if we mirror it to

Y line is the same as X, then the image is Y, X, initially X, Y becomes Y, X, for example 2017, 2016, we look at the line X is the same as Y or Y is the same as X, yes, then the image is how to change friends, absciss and the ordinate becomes 2016, 2017, yes, okay

Next, how if we mirror the original point? The original point is 0.0 If this point is a mirror, the shadow is here We pull it straight, the shadow is here, friends This is the shadow, okay? This must pass through this So this distance here is the same as the distance of the center or the original point to the shadow This is an accent

See from 4.2 we mirror to 0.0, the result is -4,-2 So how does it change? The numbers are the same, 4 and 2, only the sign changes, which here becomes negative So in general, we mirror the point A, X, Y to the line or point

0,0.o, the original point, then the image is negative x, negative y, just replace it with negative. That's the original point. Now if we mirror this point A to x = h, we take this x = 2. Okay, try it. If we make this x = 2 a mirror, then the image of point A is where? This distance is mirrored, right? 2, 2 units.

So the image is here, right? The distance must be the same. So the A-axis, the image is here, 0.2. So if we mirror point A to x = 0.2, the value that changes is the value of x. The y is still here, the y is 2, this is also 2. What changes? The change is like this.

A , we mirror it to x = h, for example, then the imagination is 2h-x, this is the change of x and the y remains y. For example, this one, at first it was 4.2, A is 4.2, we mirror it to x = 2, it means 2 is h, friends.

Then the x changes to 2 times h, which means 2 times 2 is reduced to 4. 2 times 2 is 4, then the x is reduced to 4, the y remains 2. 4 minus 4 is 0, 0.2. That's how to calculate the value of x. Now we mirror it to y = h.

For example, the mirror is y = 3, then where is the image? This is the distance of 1, so it's here, right? The image is here, the a-accent is 4.4 So the value of x from 4 remains 4, which changes is y from 2 to 4 The change is y, the method is the same as before For example, at one point, we mirror x, y towards y = h, the value of x remains

and the y changes to 2h-y, if it was 2h-x, okay, this is the change. For example, let's take another number, for example, 2.6, we mirror it to y = 1, so what about the imagination? The x remains 2, the y changes to 2xh, 2x1, minus y, 2x12, minus 6, negative 4, this is the imagination, like that.

Finally, there is something that passed earlier, y = x, now y = -x. For example, this point A, we mirror it against the line y = -x. This is the mirror. Then the shadow will be here. It will be across, here.

So the initial 4.2 we reflect towards Y = -X, then the image is -2, -4. How does that change? The position is reversed, 4 and 2, right? So it becomes 2 and 4. And the sign changes, the initial positive becomes negative. So in general, we reflect X, Y towards Y = -X, it becomes -Y, -X.

Okay, this is what you need to remember. Later we will discuss some examples of questions.

And now I will discuss how to solve a reflection using a matrix. Okay, now we will learn how to solve the problem of reflection or reflection using a matrix. If using a matrix, you must remember the matrix for each reflection. We start from reflection or reflection towards the x-axis. We have discussed earlier, if we reflect a point x, y towards the x-axis,

then the value of Y changes to negative so X, negative Y the matrix of the transformation matrix is when the position is still the same X, Y, still X, Y then use the identity 1, 0, 0, 1 here Y changes to negative negative means here so the top is for X, the bottom is for Y

Because y is negative, then negative 1 is negative here Try to the source y x, y is still x, y So we still use the identity 1, 0, 0, 1 Here the x is negative, which means the upper one, which we multiply by negative This is the matrix of transformation for the reflection of the source y Now the reflection of y is the same as x At first x, y becomes y, x This is reversed

The position is reversed, so we use the reverse identity as well. So the 1 is in the diagonal secundary, 0, 1, 1, 0. Here there is no negative, so both are positive. Now the reflection towards Y is the same as negative X, initially X, Y becomes negative Y, negative X. This is reversed, X, Y, this is Y, X. So we use the reverse identity as well.

0, 1, 1, 0, here both are negative, so negative, negative. Now, the reflection towards the original point or O, 0, 0, X, Y is still X, Y, so we still use identity. 1, 0, 0, 1, both are negative, so negative, this is negative. Now, the reflection towards X is equal to H, this is quite complicated, so I'll just give it.

This is to find the imaginary coordinate if it is reflected against x = h using a matrix. The method is like this. This is for y = h using a matrix. You can note this. How to use this matrix for reflection? Let's try the next question. Determine the imaginary coordinate of point B6.3 if it is reflected against the line y = -x.

The way the imaginary coordinate x' and y' is the same as the result of the matrix transformation with the original coordinate. Here, the reflection towards y = -x, y = -x is this one. Look at the matrix, 0 - 1, -1 0. So this matrix is what we use. For example, towards

If the value is X, it means you use this matrix. We work here.

We multiply this, we use the matrix multiplication as we have learned, the column line, 0 x 6 = 0, -1 x 3 = -3, -1 x 6 = -6, 0 x 3 = 0. Then, x' y' 0 + -3 = -3.

-6 + 0 = -6 So the imagination is the B-accent, the coordinate is -3,-6 If we use this method If we reflect a point x,y towards y = -x How does it change? y = -x, it changes, we turn and change the sign So it becomes -y,-x

So if the starting point is 6.3, we mirror it to y = -x, it turns out to be -3, -6, the result is the same, right?

Okay, we have discussed the reflection of a point, a coordinate point. Now, the reflection of a line or curve. Let's go straight to the example. Determine the equation of the line 3x + 2y - 1 = 0 if it is reflected against the line y = x. Still remember the matrix for y = x?

If we mirror the line Y = X, we just have to reverse it, right? So Y, X, if we reverse it, we use the reverse identity 0, 1, 1, 0, here both are positive, so this is also positive This is the matrix for mirroring Y = X Okay, so

x' and y' are 0, 1, 1, 0, as we have just searched times xy, we just multiply it 0 times x is 0, 1 times y is y, the top is y, the bottom is 1 times x, x plus 0 times y is 0, it means x, right? From here we get x' and y

We will return this later, initially x' = y' = x' Then the second row y' = x, we also return this to x' = y' Next, we substitute this to the initial line equation, which is 3x + 2y - 1 = 0 So we replace the x with y' and we replace the y with x'

Or the accent is not written by friends, it's okay. 3 times y the accent we throw away, so 3y 2 times x the accent is 2x. Minus 1 equals 0. This is the line of this line. Or you can reverse x first. 2x + 3y minus 1 equals 0. Okay, now I will discuss some examples of questions with different shapes.

Okay, now we discuss the first question. The point A is reflected towards the line x = 1, resulting in an a-axis with a coordinate of -1,2. The point A coordinate is, the concept is like this, for example, a point A we reflect towards a line, the image is an a-axis.

If we mirror this a-accent to the same line, then the image is a, right? So if we want to find the point a coordinate, we just mirror it, we just reflect the a-accent to this line So a-accent = -1,2 We mirror it to x = 1

X = 1 then the image is A itself if we reflect towards X = H earlier we discussed if X, Y we reflect towards X = H which changes the X only the image is how? 2H-X and Y still so this Y is still 2 which changes is the X

So the result is 3.2, is there? The answer is D

Okay, let's continue to question number 2. The triangle PQR has a coordinate P is 3.2, Q is negative 1.0, and R is negative 3.4. The triangle PQR is reflected against the line Y is equal to X, resulting in the triangle P'QR'. The coordinates P'QR' and R' are X, Y if we mirror it,

Reflect on y = x How to change? Just go back After the coordinate becomes y, x So we just have to go back So the original p is 3,2 This becomes the p-accent

2,3 Then the key -1,0 Then the key accent is 0,-1 And the R is -3,4 We reflect it to Y = X Then the R accent is 4,-3 Is there? 2,3 0,-1 And 4,-3 This one

Okay, now we continue question number 3. The result of the reflection of point P 2, -5 towards the line Y = -X, then continued reflection towards the source Y. Okay, one point X, Y if we reflect towards the line Y = -X,

Negative x, so what's the change? The position we turn and it changes the sign So this will be negative y, negative x So this p is 2, negative 5 We mirror it first towards y equals negative x will be

Negative 5 times negative is positive 5 2 times negative is negative 2 Then continue the reflection towards the source Y

towards the sum of y, which is originally x, y if we reflect it towards the sum of y then the x changes the sign, yes, it becomes negative x, y so this one, yes, we reflect it towards the sum of y the x changes the sign, this 5 becomes negative negative 5, y remains negative 2, now this is the imagination negative 5, negative 2, the answer is C okay, now we discuss question number 4

Point A -2, -4 Reflected on the line X = P Then continued reflection on the line Y = Q Resulting A double accent 6, -2 The question is the result of 3P + 4Q Point A -2, -4 We look at

Y, oh, towards X, yes, towards X = P, then what changes? The value of X, right? The X changes to 2P - X, right? 2P - -2 + 2, Y is still -4. Then we mirror again towards Y = Q.

So here what changes is the value of y, x is still 2p + 2, y changes to 2ki -4 + 4. Well, this is the picture, while in the sum the picture the coordinate is 6,-2, which means the part

Here the value is 6 and here the value is -2 So we get 2P + 2 = 6 Then 2P = 6 - 2 = 4 Then P = 4 / 2, P = 2 The key is the one next to it 2 key + 4 = -2

2 Ki = -2 -4 = -6 Then Ki is -6 divided by 2 = -3 This is P and Ki What we are looking for is 3P + 4 Ki + 4 Ki, which means 3 times P is 2 + 4 times Ki is -3

2 times 3 is 6 4 times -3 is -12 6 minus 12 is -6 So the answer is C Okay, now we discuss question number 5 Imagine the line y-x + 1 = 0 is reflected against the line y-1 = 0 Well, this is the same as y = 1 Cut the x-value at the point Okay, let's find the image first

one point x, y if we reflect it towards y = h or here h = 1, y = 1, so what changes is y so x, 2h means 2 times 1 minus y, this is the change from here we get x-accent, y-accent, friends so this part is x-accent, this part is y-accent

So we get x = x, then x = x 2 - y = y or y = 2 - y from here we get + y = 2 - y This is what we will substitute for the initial equation

Y - X + 1 = 0, what do we replace the Y? So 2 - Y' - Y' - X, the X is replaced by X' + 1 = 0, or you can throw away the accent, it's okay. So - Y - X + 1 + 2 + 3 = 0

Or we multiply it by negative, so positive y plus x minus 3 equals 0.

Or we can change this to y + x = +3 What we are looking for is the point of cut to the x-axis The point of cut to the x-axis means that y = 0 We replace y with 0 0 + x = 3 Oh, it turns out that x = 3 So if y = 0, it turns out that x = 3 So the coordinate is x,y = 3,0

The answer is D Okay, now we discuss question number 6 If the line 3x + 4y = -12 is reflected towards the sum of y Then the equation of the line of the imagination is Well, we save this first, we use it later at the end, yes Friends, remember the change if the reflection towards the sum of y If we reflect x, y

towards the source Y, then what changes is the X. So negative X, Y remains. From here we get, this is X-axis. And this part here is Y-axis. So negative X equals X-axis, or X equals negative X-axis. Y equals Y-axis, so Y doesn't change. Now we substitute the line equivalence.

3x + 4y = -12 We change the x to be -x accent 3 times -x accent + 4y we change with y accent = -12 The accent you want to throw away is also no problem 3 times -x - 3x + 4 times y = -12

Here all x are positive, so we multiply by negative. Negative 3x times negative 1 is positive 3x. 4y times negative 1 is negative 4y. And this is positive 12. Is there? 3x minus 4y equals 12. This one, 3x minus 4y equals 12.

Okay, now we discuss question number 7 or this is the last question This is the last question we will discuss in this video Line 2x-y-4=0 is reflected against the line y=x, followed by the reflection against the sum of x The result of the reflection of the line is 1 point x,y if we reflect against y=x

So what's the change? Do you remember? The change is reversed between x and y After the ordinate is reversed, it becomes y, x Then we reflect this again towards the source x If towards the source x the value of y changes, it means that y is the ordinate here, so y, this is the ordinate that changes, becomes negative

That's the change. This is the value of x-accent and this is the value of y-accent. So we get x-accent is equal to y or y is equal to x-accent. And y-accent is equal to negative x, then x is equal to negative y-accent. This is what we will substitute to the initial line equation.

2x - y - 4 = 0 We change the x to negative y-accent 2 times negative y-accent minus y, we change the y with x-accent minus 4 = 0 2 times negative y-accent minus 2y-accent minus x-accent minus 4 = 0 Or we throw away the accent, no problem

This is negative, we multiply negative all, so positive 2y + x + 4 = 0, is there? 2y + x + 4 = 0, or want x first, you can, x + 2y + 4 = 0, this one, so the answer is E.

Okay, that's it for the second transformation discussion, namely reflection See you in the next discussion, in the next material Assalamualaikum Wr Wb