Two corners of a triangle have angles of #(3 pi ) / 8 # and # pi / 8 #. If one side of the triangle has a length of #3 #, what is the longest possible perimeter of the triangle?

Question

Two corners of a triangle have angles of #(3 pi ) /  8  # and #  pi   /  8  #.  If one side of the triangle has a length of #3  #, what is the longest possible perimeter of the triangle?

Aurora Ayala · Accepted Answer

First, we note that if two angles are #alpha=pi/8# and #beta=(3pi)/8#, as the sum of the internal angles of a triangle is always #pi# the third angle is: #gamma=pi-pi/8-(3pi)/8 = pi/2#, so this is a right triangle.

To maximize the perimeter the known side must be the shorter cathetus, so it is going to be opposite the smallest angle, which is #alpha#.

The hypotenuse of the triangle will then be:

#c=a/sin alpha = 3/sin (pi/8)#

where #sin(pi/8) = sin (1/2pi/4) = sqrt((1-cos(pi/4))/2) =sqrt((1-sqrt(2)/2)/2)#

#c= (3sqrt(2))/sqrt(1-sqrt(2)/2)#

while the other cathetus is:

#b = a/tan(pi/8)#

where #tan(pi/8) = sqrt((1-sqrt(2)/2)/(1+sqrt(2)/2))#

#b=3sqrt((1+sqrt(2)/2)/(1-sqrt(2)/2))#

Finally:

#a+b+c = 3+ (3sqrt(2))/sqrt(1-sqrt(2)/2)+3sqrt((1+sqrt(2)/2)/(1-sqrt(2)/2))#

Claire Akin · Answer

undefined To find the longest possible perimeter of the triangle, we need to consider the triangle inequality theorem, which states that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given that two angles of the triangle are $ \frac{3\pi}{8} $ and $ \frac{\pi}{8} $, the remaining angle can be found by subtracting the sum of the given angles from $ \pi $ radians.

Remaining angle = $ \pi - \left(\frac{3\pi}{8} + \frac{\pi}{8}\right) = \pi - \frac{4\pi}{8} = \pi - \frac{\pi}{2} = \frac{\pi}{2} $

Now, we know that the third angle of the triangle is $ \frac{\pi}{2} $, making it a right triangle.

Given one side has a length of 3 units, we can use trigonometric ratios to find the lengths of the other sides. In a right triangle, the lengths of the sides are related by the trigonometric functions sine, cosine, and tangent.

Let's use the side with length 3 as the side opposite the angle $ \frac{\pi}{8} $ and use the tangent function:

$$ \tan\left(\frac{\pi}{8}\right) = \frac{x}{3} $$

$$ x = 3 \tan\left(\frac{\pi}{8}\right) $$

Now, we can use the Pythagorean theorem to find the length of the third side:

$$ \text{Third side} = \sqrt{3^2 + (3 \tan(\frac{\pi}{8}))^2} $$

With the length of the third side determined, we can calculate the perimeter of the triangle:

$$ \text{Perimeter} = 3 + 3 + \text{Third side} $$

This gives us the longest possible perimeter of the triangle.