A triangle has two corners with angles of # pi / 2 # and # ( pi )/ 8 #. If one side of the triangle has a length of #13 #, what is the largest possible area of the triangle?

Question

A triangle has two corners with angles of #  pi  /  2  # and # ( pi  )/  8   #.  If one side of the triangle has a length of #13  #, what is the largest possible area of the triangle?

Sadie Adams · Accepted Answer

The area is #169/(2(sqrt(2)-1))~~204#

If one of the angles of the triangle is #pi/2#, it is a right triangle. The measure of the third angle of the triangle must be #pi/2-pi/8=(3pi)/8#. The shortest side of this right triangle will be the leg that is opposite the angle measuring #pi/8#. Let #x#= the length of the longer leg. The area, #A#, of the triangle will be

#A=13/2x#

From trigonmetry

#x=13/tan(pi/8)#

So

#A=13^2/(2tan(pi/8))=169/(2(sqrt(2)-1))~~204#

#A=13/2x#

From trigonmetry

#x=13/tan(pi/8)#

So

#A=13^2/(2tan(pi/8))=169/(2(sqrt(2)-1))~~204#

Noah Atkinson · Answer

undefined To find the largest possible area of the triangle given two angles and one side length, we can use trigonometry.

Let's denote the angles of the triangle as $ \frac{\pi}{2} $, $ \frac{\pi}{8} $, and $ \theta $ (where $ \theta $ is the remaining angle).

Using the fact that the sum of angles in a triangle is $ \pi $, we find $ \theta = \pi - \left(\frac{\pi}{2} + \frac{\pi}{8}\right) = \frac{3\pi}{8} $.

Now, we can use the Law of Sines to find the lengths of the other two sides of the triangle.

$$ \frac{a}{\sin(\frac{\pi}{2})} = \frac{13}{\sin(\frac{3\pi}{8})} $$
$$ a = 13 \cdot \sin(\frac{\pi}{2}) \cdot \csc(\frac{3\pi}{8}) $$
$$ a \approx 18.44 $$

$$ \frac{b}{\sin(\frac{\pi}{8})} = \frac{13}{\sin(\frac{3\pi}{8})} $$
$$ b = 13 \cdot \sin(\frac{\pi}{8}) \cdot \csc(\frac{3\pi}{8}) $$
$$ b \approx 35.89 $$

Now, we can find the area of the triangle using the formula $ \frac{1}{2} \cdot a \cdot b $.

$$ \text{Area} = \frac{1}{2} \cdot 18.44 \cdot 35.89 $$
$$ \text{Area} \approx 331.42 $$

So, the largest possible area of the triangle is approximately $ 331.42 $ square units.