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Reflection
Reflection in Quants? Isn’t this a physics concept we learned back in school?
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.
Reflection on the x-axis
reflection1.png [ 36.26 KiB | Viewed 1772 times ]
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).
reflection2.png [ 36.53 KiB | Viewed 1780 times ]
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.
reflection3.png [ 36.22 KiB | Viewed 1743 times ]
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).
reflection4.png [ 36.33 KiB | Viewed 1733 times ]
Reflection on the Origin
If we want to reflect the same point (2, 3) about the origin, then that point will be (-2, -3)
Untitled.png [ 61.11 KiB | Viewed 1729 times ]
Another example can be the point (1, -2) which if we reflect on the origin, we get (-1, 2).
reflection5.png [ 35.96 KiB | Viewed 1715 times ]
Reflection on the line X = Y
This is another case, that we can encounter while solving coordinate geometry question
reflection7.png [ 42.38 KiB | Viewed 1703 times ]
Reflection on 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.
reflection8.png [ 42.68 KiB | Viewed 1695 times ]
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)
Attachment:
reflection1.png [ 36.26 KiB | Viewed 1772 times ]
- 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).
Attachment:
reflection2.png [ 36.53 KiB | Viewed 1780 times ]
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.
Attachment:
reflection3.png [ 36.22 KiB | Viewed 1743 times ]
- 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).
Attachment:
reflection4.png [ 36.33 KiB | Viewed 1733 times ]
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
Attachment:
Untitled.png [ 61.11 KiB | Viewed 1729 times ]
Another example can be the point (1, -2) which if we reflect on the origin, we get (-1, 2).
Attachment:
reflection5.png [ 35.96 KiB | Viewed 1715 times ]
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).
Attachment:
reflection7.png [ 42.38 KiB | Viewed 1703 times ]
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 on 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)
Attachment:
reflection8.png [ 42.68 KiB | Viewed 1695 times ]
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