Reflection of a point on Coordinate Plane

Reflection in Quants?  Isn’t this a physics concept we learned back in school?  

• Well, yes, it is.  However, a similar topic, which works on the same principle, is present in mathematics too.

Reflection in mathematics or quant takes place in the cartesian or coordinate plane and not in real-life space. Let us discuss this in detail.  

Reflection of a Point 

Most of the questions that come in GMAT are for a point or a line.

In this article, we are going to cover some advanced concepts of coordinate geometry, starting with the reflection of a point.

• the x-axis,  
• the y-axis,  
• the origin,   
• the lines x = y,  
• and x = – y  

Reflection on the x-axis  

• Let us take a point A (2, 3) 
• Now, if we reflect this point about the x-axis then it means the point will go to the other side of the x-axis (shown in the diagram)   

• Remember that the reflected point (A’ in this case) and the original point (A in this case) are always equidistant from the line about which the reflection is taken (x-axis in this case).

• Let us take another point say (-3, 2), now, if we reflect this point on the x-axis, then notice that once again the sign of the y coordinate will change and the coordinate of the reflected point will be (-3, -2).

Thus, if the point (x, y) is reflected about the x-axis, then the co-ordinate of the reflected point is (x, -y)

Reflection on the y-axis  

If want to reflect the same point (2,3) about the y-axis, then the reflected point will be (-2,3)  

See, in this case, the value of the y-co-ordinate does not change.  

• It is the sign of the x coordinate that is changing because it is moving from the positive side of the x-axis to the negative side.  

Let us take another point (3, -2). If we reflect this point about the y-axis, we’ll get the reflected point as (-3, -2).

Hence, in this case, we can say if the point (x, y) is being reflected about the y-axis, then the co-ordinate of the reflected point will be (-x, y)

Reflection on the origin  

If we want to reflect the same point (2, 3) about the origin, then that point will be (-2, -3)  

• In this case the sign of both x and y coordinate changes  

Another example can be the point (1, -2) which if we reflect on the origin, we get (-1, 2).

Hence, in this case, we can say if the point (x, y) is being reflected about the origin, then the co-ordinate of the reflected point will be (-x, -y)  

Reflection on the line X = Y  

This is another case, that we can encounter while solving coordinate geometry question  

• We know that the x = y line passes through the origin and makes an acute angle (45° precisely) with the positive side of the x-axis.

• Now if we take a point, say (2, 3), and reflect it about the line x = y, then the coordinate of the reflected point will just get reversed and it becomes (3, 2).

So, if we have point (x, y) and we reflect it about the line x = y, then in that case, we get the reflected point as (y, x) 

Reflection about the line X = – Y  

We know that the x = -y line also passes through the origin but the only difference is that it has a negative slope.

• This means that it makes an obtuse angle (135° precisely) with the positive side of the x-axis  

• If we take the point (2, 3) and reflect it about the line x = -y, then the coordinates that we’ll get will be reversed in value and signs.  Here the reflected point will be (-3, -2)  

All we need to remember is that if we have point (x, y) and we reflect it about the line y = -x, then in that case, we get the reflected point as (-y, -x)