Here's another trig question

[cllynn213]

In a rhomboid the obtuse angle is 123 degrees, the longer side is 50 mm and the longer diagonal is 80 mm. Find the shorter side of the rhomboid.

Can someone please help? Thanks!

 

[Gene]

Draw the right triangle with the shorter side as the hypot and the base the line from the obtuse angle to the right angle = x. The other leg is h.
tan(180-123)=h/x
h²+(50+x)²=80²
Two equations intwo unknowns to be solved for h and x.
the shorter side = sqrt(h²+x²)

 

[soroban]

Hello, cllynn213!

In a rhomboid the obtuse angle is \(\displaystyle 123^o\), the longer side is \(\displaystyle 50\) mm, and the longer diagonal is \(\displaystyle 80\) mm.
Find the shorter side of the rhomboid

 B
  *
   *   *  80
  x *         *
     *.123°         * 
      * * * * * * * * * * *
      A        50         C

We are given: \(\displaystyle \,A\,=\,123^o,\:AC\,=\,50,\:BC\,=\,80\)
\(\displaystyle \;\;\)and we want: \(\displaystyle x\,=\,AB\).


Using the Law of Sines:
\(\displaystyle \L\;\;\;\frac{\sin B}{50}\,=\,\frac{\sin123^o}{80}\)\(\displaystyle \;\;\Rightarrow\;\;\sin B\,=\,0.524169105\;\;\Rightarrow\;\;B\,\approx\,31.6^o\)

Then: \(\displaystyle \,C\:=\:180^o\,-\,123^o\,-\,31.6^o\:=\:25.4^o\)

Hence: \(\displaystyle \L\,\frac{x}{\sin25.4^o}\,=\,\frac{80}{\sin123^o}\)\(\displaystyle \;\;\Rightarrow\;\;x\,\approx\,40.9\)


Using the Law of Cosines:
\(\displaystyle \;\;\;80^2\:=\:x^2\,+\,50^2\,-\,2\cdot x\cdot50\cdot\cos123^o\)

We have the quadratic: \(\displaystyle \,x^2\,+\,54.46x\,-\,3900\:=\:0\)

Quadratic Formula: \(\displaystyle \L\,x\:=\:\frac{-54.46\,\pm\,\sqrt{54.46^2\,+\,4(3900)}}{2(1)} \:= \:\frac{-54.46\,\pm\,136.2567}{2}\)

\(\displaystyle \;\;\;\)Therefore: \(\displaystyle \,x\,\approx\,40.9\)



*

 

[cllynn213]

I had no idea how to even get started but the advice you guys gave was tops. Thanks!

 

[Denis]

Or a lazy guy like me would jump right in the Cosine Law:

x = shorter side
k = cos(123)

80^2 = x^2 + 50^2 - 2(x)(50)k
6400 = x^2 + 2500 - 100kx
x^2 - 100kx - 3900 = 0
Solve to get x = ~40.8972