Kind of confused: finding points giving required angles

M

Maryanne

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Joined
Jan 11, 2006
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Hi, I'm a little confused by this problem and was wondering if someone could just get me started. Any help would be much appreciated.

The manufacturers of a reclining lawn chair would like to have the chair positioned at the following angles: 95, 120, 135, 160, and 175 (degrees).

The lawn chair is shape like a trianlge with point A, B and C. AC is 65 cm and AB is 45 cm.

Find the positions for the notches on BC that will produce the required angles. Give a complete solution.
 
Maryanne said:
Yeah, i know but I don't have a scanner!
Click to expand...
So you'll need to describe the picture explicitly.

Thank you.

Eliz.
 
The diagram is of a triangle. point A is on the left, point B is on the top and point C is on the right. BC is the back of the lawn chair that is moved accordingly. Therefore point A is where the person would sit. AC is 65 cm, and AB is 45 cm.
 
I see. What you need most is the idea that the measure of the INTERIOR angle is calculated as 180º less the EXTERIOR angle that you are given. This gives you a unique triangle, SAS, so you should be able to solve for the third side quite easily.
 
So let me see if I understand this, I would solve by using the cosine law; two sides (AC=65 cm and AB=45cm) and one of the angles given? Which means I would be trying to find the unknown side of BC........and the I guess I would do this five times ( because five different angles were given).
 
You are SO close! You cannot use the given angles directly. They are not angles of the triangle. You need:

180º - 95º = 85º
180º - 120º = 60º
etc.

When you sit on A and lean your head back on the pillow at B, the angles given will be correct EXTERNAL angles.
 
Hello, Maryanne!

I don't understand the problem . . .

The manufacturers of a reclining lawn chair would like to have the chair positioned
at the following angles: 95, 120, 135, 160, and 175°.

The lawn chair is shape like a triangle with point A, B and C.
AC is 65 cm and AB is 45 cm.

Find the positions for the notches on BC that will produce the required angles.
Give a complete solution.
Click to expand...
Code: 
                      B
                      *
                     /  \
                    /     \
                45 /        \ a
                  /           \
                 /              \
      D - - - - * - - - - - - - - *
                A       65        C
I assume the person sits in angle BAD, which will have the five given values.

But I don't understand how notches are placed on BC to adjust the angle.


I suspect that the diagram looks like this:
Code: 
                      B
                      *
                     /  \
                    /     \
                 45/        \65
                  /           \
                 / θ            \
      D - - - - * - - - - - - - - *
                A        b        C
and the notches are placed on AC.


Assuming I'm correct, let's try the first angle: \(\displaystyle \,\angle BAD\,=\,95^o\)
\(\displaystyle \;\;\)Then \(\displaystyle \theta\,=\,\angle BAC\,=\,85^o\)

Using the Law of Sines, we can find \(\displaystyle \angle C:\;\;\frac{\sin C}{45} \:=\:\frac{\sin85^o}{65}\)

Then: \(\displaystyle \,\sin C\,=\,\frac{45\cdot\sin85^o}{65}\:=\:0.689673253\;\;\Rightarrow\;\;C\:\approx\;51.0^o\)

Hence: \(\displaystyle \,\angle B \:=\:180^o\,-\,85^o\,-\,51^o\:=\:44^o\)


Then: \(\displaystyle \,\frac{b}{\sin 44^o}\,=\,\frac{65}{\sin 85^o}\;\;\Rightarrow\;\;b\:=\:\frac{65\cdot\sin44^o}{\sin85^o} \:=\:45.42527042\)

Therefore, for \(\displaystyle \angle BAD\,=\,95^o:\;b\,=\,AC\,\approx\,45.4\) cm.
 
actually AB is 45 cm and AC is 65 cm. So, BC is the unknown that's been solved for. But I think that you method is correct Soroban
 
soroban said:
Code: 
                      B
                      *
                     /  \
                    /     \
                45 /        \ a
                  /           \
                 /              \
      D - - - - * - - - - - - - - *
                A       65        C
                                     \
                                      \
                                       \
                                         \F
Click to expand...
The structure rotates at A and BF slides at C
 
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