the point (2, –4) is reflected across the line y = –1. what are the coordinates of the image? This is a topic that many people are looking for. bluevelvetrestaurant.com is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, bluevelvetrestaurant.com would like to introduce to you Image Point Coordinates After Reflection Across any Line. Following along are instructions in the video below:
Kumar. Let me first thank my subscriber and i really appreciate the request. The question question here is regarding reflection across a line.
Ill put this under the category thinking questions. Tips. The question.
Here is determine the coordinates of the image for a point p 3. 4. Which is reflected in the mirror.
Placed along the straight line. Y. Equals to minus.
2x plus. 3. Now you can pause the video copy the question attempt and then look into my suggestions.
We already have a couple of videos who have did we have discussed what happens when we reflect or when we use mirror alone horizontal vertical or diagonal on a coordinate plane now here it is a line which is not along the axis so it makes things slightly more. Difficult so what we have here is 034. So and we have a line.
Which is minus. Y equals. 2.
Minus. 2x plus. 6.
That means let us say. This is 6 for us the y intercept and slope is minus 2. So that means.
It is going downwards and this x intercept will be at 3 right then it gives you slope of minus 2. Let. Me first sketch a line here so let this be the line.
Which is what y intercept or 6. And if this point is 3. You know the slope is minus 2.
So that is the line for us. Well call this line as y equals. 2.
So let me say. This is y axis. This is x axis.
Okay. Now we are given a point 3 4. So approximately so normally as you find my videos.
We always use very approximate sketches. Which are good enough to give us just the right results. So so we say.
3 4. Is this point p. Alright.
So p. Is 3. Comma.
4. For us. Writes x value.
3. Y value. Slightly more than half of this.
Ok. So. Thats the origin for us.
Now. Lets read the question once again we need to determine the coordinates of the image for a point p. Which is here.
Which is reflected in the mirror placed along the straight line. Y equals. 2.
Minus. 2x plus. 6.
So this is the mirror for you so. Does the reflected means it comes somewhere on this side is it okay. So that is how the reflection is going to take place so find the image point coordinates.
Now how can you do that one thing. Which we all understand is that this mirror image is going to be had a distance. Which is seen as in front of it right so.
So that is the concept which we are going to utilize let us see that from p. We find a point on this line. Which is so that it just makes a perpendicular to the line.
So. Let me call this point. As q.
Okay. So we know pq is perpendicular to the line. Lets say line is l4.
So this is the line for us now. We need to find this point. Q.
On the line that will help us to get the image point. Which is as far behind q. As p.
Is in front of it you get an idea. So we are looking for a point p. This side and these two distances to be same i hope you got the concept correct right so what is the equation of the line.
Which is perpendicular to the given line l. So we say l. Which is perpendicular will be y equals.
2. Slope should be negative reciprocal right so slope in our case will be instead of minus 2. It is going to be 1 2.
So it is harvicks and we dont know what beers will right be here now since this line passes through the point 3 4. Well substitute. 3 for x and 4 for y.
So. 4. Equals to 1 2.
Times. 3 plus b. And from here.
We can find what b. Is it is 4 minus. 3.
Over 2 and that is the value of b. So. Which is 8 minus.
Let me write. 8 minus. 3.
Over. 2 is the value of b. Right or we can say b is equal to 5 over 2 right so b is 5 over 2.
So that gives us the equation of a perpendicular line. So let me write down. Here.
Now perpendicular line will be y equals to 5 half of x and then y there is a b. Y intercept is 5 over 2 or 2 point 5 5. So that becomes the equation of the line to write it in a simpler form we can just multiply by 2 all these values so we get 2y or okay.
So 2y equals 2x plus. 5. So that becomes the equation of this particular line right.
Now. We have two lines. So line.
1 as y goes to minus. 2x plus. 6.
This line second. One is to y equals. 2x plus.
5. So that is the second line. Now we need to find point of intersection.
Which we could find using elimination or substitution let me write down. What x is here. So we can say x is 2.
Y minus. 5. So well substitute this value of x.
In the given equation. So. We say y equals.
2. Minus. 2 x.
So. 2y minus. 5.
Thats the value of. X. 6.
Right. So. That gives us.
Minus 4. Y. Plus.
10. Plus. 6.
Equals. 2y. Bringing.
It to the left. Side. Gives.
Us. 5. 5.
Equals to 16 or y equals to 16 or 5. So we get the y value of 16 over 5. Once we know the y value.
We can find the value of x. Right. So.
What is x. Equals. 2.
X. Is equals to 2 y. Time minus 5.
So 2 times. 16. Over.
5 minus. 5. That is what we have so that is 32 minus.
5. Times. 5.
Is 25 divided by 5. Oh. That is 7 over.
5. So we get these coordinates for q. So let me write down the coordinates for q.
Now so the corners for qr x values. 7. Over 5 and the y value is 16 over.
5. So that is the coordinate of q. Now the difficult part of the question is to find the corners of p dash.
Which is on the other side of q. How do you find that now. We know q is the midpoint since q.
Is the midpoint and we dont know what coordinates of p. Are let us assume that the coordinates of p. Dash are x.
And y right. Since q. Is the midpoint in that case.
X. Plus. 3.
Divided by 2 should be equals to the coordinates x. Value of q. Which is 7 over.
5 right and the y value should be y plus. 4. Divided by 2 should be equal to the y value.
Which is 16 over. 5. Do you get an idea right so.
What weve done here is we know q is a midpoint. So q is midpoint since q. Is midpoint x.
Midpoint of p. Dash and p. Which is 3 comma.
4. So x plus. 3.
Divided by 2 should be goes to the q value. Which is 7 over 5 let me write here 7 over 5 and 16 over 5 connie so we get two different equations and these equations will give us a value of x and y. So lets cross multiply and then find so we have x equals.
2. When you cross multiply you get 14 over 5. Then i never take away.
3 minus 3. So we have 14 minus 15. Over 5.
Which is minus 1. Over. 5.
That is the x value. And as far as the y. That is concerned will have y plus.
4. Equals. You cross multiply.
32. Over. 5 or y.
Is equals to 32 over. 5 minus. 4.
Which is 20 right. 32. Minus.
20. Over. 5.
That is equals to 12 over 5. So 12 over 5 is the y value right so approximately that is i mean im not writing approximately. We are writing in fractions itself.
So so what we get here is the point slightly on the left side of the y axis. So we say image is p. Dash.
Which is x well is minus 1. Over. 5 and y is 12 over 5.
So that becomes our answer you get an idea so. The steps involved are first point 3 4. Will lie on a line.
Which is perpendicular to given line right so this line is perpendicular slope of perpendicular line is negative reciprocal. So we got the slope as 1 2. Now find b using the given point.
3. And 4. Once you get the equation of the perpendicular line.
Now we need to find intersection of these two points right so we found at this stage. We are trying to find intersection right so intersection of the two given lines these two lines creek. That point of intersection is actually the midpoint between the point p.
Hands image right now. So once you find the center point q. Then which we found as 7 over.
5. That is a midpoint right then use the midpoint formula to find the image points. So we found the image point as minus 1.
Over. 5 is the x value and 12 over. 5 is the y value right so that is how we do it i hope you understand and appreciate the steps.
Well take few more examples to help you develop this concept. Thank you and all the best. .
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