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I am doing the $2016$ CIMC (Canadian Intermediate Mathematics Contest) contest and I am kind confused with the question A6 meaning.

I just wonder what does the line is reflected in the line mean?

The line is reflected in the line with equation $x + y = 1$.

Here is the whole question: A line has equation $y=kx$,where $k$ not equal $0$ and $k$ not equal $−1$. The line is reflected in the line with equation $x + y = 1$. Determine the slope and the y-intercept of the resulting line, in terms of $k$.

From CIMC $2016$ part a 6.

Shawn
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    Are you confused because you are not familiar with reflections? Or because you are not sure what reflection is being applied to what in this problem? – Misha Lavrov Nov 15 '21 at 03:34
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    If someone asked you for the reflection of the point $(3,0)$ in the line with equation $x+y=1,$ could you answer that question? (This is another way of asking whether you are familiar with what a reflection in a line is.) – David K Nov 15 '21 at 03:40
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    BTW, this is from 2016. – hyper-neutrino Nov 15 '21 at 03:41
  • Think of y=kx as a ray of light and x+y=1 as a mirror. Light falls on the mirror and gets reflected. So I think that you’re supposed to find the equation of the reflected ray of light using the law of reflection, which states that angle of incidence =angle of reflection. – Koro Nov 15 '21 at 03:43
  • "a line is reflected in the line" means "a line is reflected over the line", https://flexbooks.ck12.org/cbook/ck-12-interactive-geometry-for-ccss/section/2.4/primary/lesson/reflections-geo-ccss/ – markvs Nov 15 '21 at 03:50
  • Seconding @hyper-neutrino's comment: OP, please edit your question to reflect that the problem is A6 on the 2016 CIMC, not the 2014 CIMC. – Brian Tung Nov 15 '21 at 07:48
  • The original problem can be found here: https://www.cemc.uwaterloo.ca/contests/past_contests/2016/2016CIMC.pdf – Brian Tung Nov 15 '21 at 07:49
  • @MishaLavrov I am just confused with the property of reflection between lines. – Shawn Nov 15 '21 at 15:09
  • @DavidK, maybe no. I don't know what is the reflection between 2 lines. – Shawn Nov 15 '21 at 15:10
  • @Koro,I just checked the solution and it using reflection line is perpendicular to another line this property to solve the question. I just confused why prependicular has this property. – Shawn Nov 15 '21 at 15:21
  • There is only one time "perpendicular" is mentioned in the solution. This is in the context of one step which asks for the reflection of the single point $O=(0,0)$ in the line $x+y=1.$ So we are (at that moment) not concerned with two lines, only one point that is to be reflected in one line. The image of point $O$ reflected in the line $x+y=1$ is $Q$. Now, never mind what the published solution said next; how would you find the coordinates of the point $Q$? – David K Nov 15 '21 at 18:08
  • @DavidK, I totally understand what you mean, but I believe that there is no way for us really find the coordinates of the point because one line use equation with variable to present (y = kx). – Shawn Nov 15 '21 at 23:15
  • The way to show you understand is to find the coordinates. On the other hand, the first step to gaining understanding is to recognize that you do not understand something. To reflect any geometric figure (such as a line) "in" or "over" a line $L$ means that you reflect every point of the figure over the line $L$. So understanding the reflection of a point over a line is essential to understanding reflection of anything else over a line. – David K Nov 16 '21 at 03:55
  • @DavidK I finally get the solution now based on your explanation of "in". Thanks so much for the classification! – Shawn Nov 16 '21 at 04:29

2 Answers2

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Let L1 and L2 be two lines, which should intersect. Reflection of L1 on L2 refers to a new line L1' such that

for the angle L1-L1' the line L2 will be the bisector (angle of incidence = angle of reflection, as L2 plays the role of the mirror)

P Vanchinathan
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  • I understand the definition of reflection now. I just checked the solution and it using reflection line is perpendicular to another line this property to solve the question. I just confused why prependicular has this property. – Shawn Nov 15 '21 at 15:23
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Good. Now you that you understood it, a sketch as to what happens to a $15^{\circ}$ incident ray is given so that you can verify your analytical result with the numerical data you obtained.

enter image description here

Narasimham
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