Two corners of a triangle have angles of #(5 pi ) / 12 # and # ( pi ) / 8 #. If one side of the triangle has a length of #4 #, what is the longest possible perimeter of the triangle?

Answer 1

I will let you do the final calculation.

Sometimes a quick sketch helps in the understanding of the problem. That is the case hear. You only need to approximate the two given angles.

It is immediately obvious (in this case) that the shortest length is AC.

So if we set this to the given permitted length of 4 then the other two are at their maximum.

The most straight forward relationship to use is the sine rule.

#(AC)/sin(B)=(AB)/sin(C)=(BC)/sin(A)# giving:

#(4)/sin(pi/8)=(AB)/sin((5pi)/12)=(BC)/sin(A)#

We start be determining the angle A

Known: #/_A+/_B+/_C=pi" radians"=180 #

#/_A+pi/8+(5pi)/12=pi" radians"#

#/_A=11/24 pi" radians" -> 82 1/2" degrees"#

This gives:

#color(brown)((4)/sin(pi/8)=(AB)/sin((5pi)/12)=(BC)/sin((11pi)/24))#

Thus #AB=(4sin((5pi)/12)) /sin(pi/8)#

and #BC=(4sin( (11pi)/24))/sin(pi/8) #

Work these out and add then all up including the given length of 4

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Answer 2

#24.459#

Let in #\Delta ABC#, #\angle A={5\pi}/12#, #\angle B=\pi/8# hence
#\angle C=\pi-\angle A-\angle B#
#=\pi-{5\pi}/12-\pi/8#
#={11\pi}/24#
For maximum perimeter of triangle , we must consider the given side of length #4# is smallest i.e. side #b=4# is opposite to the smallest angle #\angle B={\pi}/8#
Now, using Sine rule in #\Delta ABC# as follows
#\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}#
#\frac{a}{\sin ({5\pi}/12)}=\frac{4}{\sin (\pi/8)}=\frac{c}{\sin ({11\pi}/24)}#
#a=\frac{4\sin ({5\pi}/12)}{\sin (\pi/8)}#
#a=10.096# &
#c=\frac{4\sin ({11\pi}/24)}{\sin (\pi/8)}#
#c=10.363#
hence, the maximum possible perimeter of the #\triangle ABC # is given as
#a+b+c#
#=10.096+4+10.363#
#=24.459#
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Answer 3

To find the longest possible perimeter of the triangle, we need to determine the length of the other two sides. We can use the Law of Sines to find the lengths of these sides.

Let's denote the angle with measure ( \frac{5\pi}{12} ) as ( A ) and the angle with measure ( \frac{\pi}{8} ) as ( B ). Let ( C ) be the third angle of the triangle.

We know that the sum of angles in a triangle is ( \pi ), so we can find ( C ) by subtracting the sum of angles ( A ) and ( B ) from ( \pi ).

( C = \pi - \left(\frac{5\pi}{12} + \frac{\pi}{8}\right) )

Now, knowing all three angles of the triangle, we can use the Law of Sines to find the lengths of the sides. Let ( a = 4 ) be the length of the side opposite angle ( A ).

( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} )

We can solve this equation to find the lengths ( b ) and ( c ).

Once we have the lengths of all three sides, we can calculate the perimeter of the triangle as the sum of the lengths of its sides. Then, we can determine the longest possible perimeter.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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