The height TR of a tree

eddy2017

eddy2017

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Hi, dear tutors and helpers:

I need help with this. I'd appreciate if you can send me links for tutorials and/or tell me what to study to start analyzing this. I have already studied intersecting lines, secant lines, etc, but nothing is helping. someone suggested the theorem of intersecting lines, but I have not found that online. maybe free academy has a video about that.



The height TR of a tree may be measured by using similar triangles. A mirror is placed at point M so that the top of the tree is sighted in the mirror by a person standing at point P. The person's eye is at point E. Given the measurements shown in the diagram what is the length of TR?
I am attaching a pic that comes with the problem
 

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Forgot to add that TR has a bar on top that I know it means it is a line segment.
 
eddy2017 said:
Hi, dear tutors and helpers:

I need help with this. I'd appreciate if you can send me links for tutorials and/or tell me what to study to start analyzing this. I have already studied intersecting lines, secant lines, etc, but nothing is helping. someone suggested the theorem of intersecting lines, but I have not found that online. maybe free academy has a video about that.



The height TR of a tree may be measured by using similar triangles. A mirror is placed at point M so that the top of the tree is sighted in the mirror by a person standing at point P. The person's eye is at point E. Given the measurements shown in the diagram what is the length of TR?
I am attaching a pic that comes with the problem
Click to expand...
What is the FIND of the problem?

Do you know the law of reflection? (If not Google it.
 
Eddy

The law of reflection is a law of physics rather than mathematics. It will tell you that two angles have equal measure. No need to study it deeply. The mathematics that you need to know to do this problem are theorems on similar triangles and how to use algebra to solve ratio problems. Focus on the math.
 
That is a good tip ?. Thank you Jeff. I'll go there immediately
 
JeffM said:
The law of reflection is a law of physics rather than mathematics.
Click to expand...
However for this problem, the law of reflection is needed to establish why and which triangles are similar.
 
Subhotosh Khan said:
However for this problem, the law of reflection is needed to establish why and which triangles are similar.
Click to expand...
I will study both topics. Give some time and I will come back with something
This is the joy of discovering Math. And having minds like you to fall back to in case doubt sets in. And it will.
 
JeffM said:
Eddy

The law of reflection is a law of physics rather than mathematics. It will tell you that two angles have equal measure. No need to study it deeply. The mathematics that you need to know to do this problem are theorems on similar triangles and how to use algebra to solve ratio problems. Focus on the math.
Click to expand...
Yes, the angle of incidence is equal to the angle of reflexion. Not much but it's more than I had. Keep on keeping on!
 
eddy2017 said:
Yes, the angle of incidence is equal to the angle of reflexion. Not much but it's more than I had. Keep on keeping on!
Click to expand...
Correct!

In the given figure in OP - which angle is angle of incidence ?

which angle is angle of reflection?​
 
I got home 20 minutes ago. Left all my notes about the two laws of reflection back on my desk at work. I'll answer tomorrow first thing in the morn. Forever grateful for your help. Both you and Jeff.
 
Sorry, busy day.

the angle of incidence: angle formed when the ray reaches the point of reflection (M) angle formed by MRT
angle of reflection or r formed by TME.
 

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Eddy

You are started down the right path: you have done some good things. Diagram. Excellent. Great way to start. Then you label an angle as [imath]a_i[/imath], but you do so implicitly. Always label the unknown numbers before you start writing equations and so on. But it is best to do so explicitly. Write down [imath]a_i = \angle \text { of incidence}[/imath] if that is what you mean.

But you failed to describe the angle in your diagram associated with [imath]a_i[/imath], which is RMT rather than MRT. Moreover, you failed to give a label to the angle of reflection. It is not required to be consistent in your notation, but it helps a lot. So you should have labeled [imath]a_r = \angle \text {of reflection}[/imath] and indicated that on your diagram.

Unfortunately, you are wrong as to the angle that is the angle of incidence and wrong as to the angle of reflection.

Do you know what the "normal" means? The angles of incidence and reflection are defined relative to the normal. Look at the first diagram at

en.wikipedia.org

Reflection (physics) - Wikipedia

en.wikipedia.org en.wikipedia.org
 
The normal is the perpendicular line that divides both the angle of incidence and the angle of reflection. It goes straight up from the point of incidence.
 
eddy2017 said:
The normal is the perpendicular line that divides both the angle of incidence and the angle of reflection. It goes straight up from the point of incidence.
Click to expand...
Where is it in your diagram?
 
Now I know the angles are wrong but this is the first diagram rectified.
 

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now lt me check this
'Unfortunately, you are wrong as to the angle that is the angle of incidence and wrong as to the angle of reflection.'

watching this video .it is good.https://www.youtube.com/watch?v=yscrBkfPRyc
 
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OK. You have the normal. Put an end point N on it (for normal) so you can describe the angles that you are talking about.

Now you are TECHNICALLY wrong as to the angle of incidence and as to the angle of reflection, but the shaded angles are indeed the ones that you are ULTIMATELY interested in mathematically.

What you have labeled [imath]a_i[/imath] and [imath]a_r[/imath] are of importance even though they are not the angles of incidence and reflection. I suggest that go

[math]c_i \text { is the measure of } \angle \text { TMR, and }\\ c_r \text { is the measure of } \angle \text {EMP .}[/math]
Now what are the angle of incidence and the angle of reflection?

I looked at the video. It is good. I agree. But it is focused on a harder problem than this one.

Nevertheless, it gets at my point. The angles that you are talking about are complements of the angles of incidence and reflection. Did you notice that early in the video?
 
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