Unknown triangle side length

[miamivince]

A triangle has 2 sides, 112 and 120 cms. Point c in the triangle lies 65 cm from each vertex. Find the length of the third side ? Is this middle point a centroid or orthocenter ? Is there a formula we can use ? Thanks.

 

[soroban]

Hello, miamivince!

Triangle \(\displaystyle ABC\) has two sides,\(\displaystyle AB = 112,\:AC = 120\) cm.
Point \(\displaystyle D\) in the triangle lies \(\displaystyle 65\) cm from each vertex.
Find the length of the third side.

I have a rather long game plan.
I'm sure there is a shorter solution.

                A
                o
               /*\
              / * \
             /  *65\
        112 /   *   \ 120
           /  α o β  \
          /   * D *   \
         /  *   θ   *  \
        / *65       65* \
       /*               *\
    B o - - - - - - - - - o C
                x

We have \(\displaystyle \Delta ABC\) with \(\displaystyle AB = 112,\:AC = 120.\)
We have point \(\displaystyle D\) with \(\displaystyle AD = BD = CD - 65.\)
Let \(\displaystyle \alpha = \angle ADB,\;\beta = \angle ADC,\:\theta = \angle BDC.\)

Law of Cosines: .\(\displaystyle \cos\alpha \:=\:\dfrac{65^2 + 65^2 - 112^2}{2(65)(65)} \:=\:-0.704142012\)
. . . . . . . . . \(\displaystyle \alpha \:\approx\:134.76^o\)

Law of Cosines: .\(\displaystyle \cos\beta \:=\:\dfrac{65^2 + 65^2 - 120^2}{2(65)(65)} \:=\:-0.484497041\)
. . . . . . . . . \(\displaystyle \beta \:\approx\:118.98^o\)

Hence: .\(\displaystyle \theta \:=\:360^o - 134.76^o - 118.98^o \:\:106.26^o\)


Law of Cosines: .\(\displaystyle x \;=\;65^2 + 65^2 - 2(65)(65)\cos106.26^o \;=\;10815.97102\)

Therefore: .\(\displaystyle x \:=\:103.9998607 \:\approx\:104.0\)

 

[Deleted member 4993]

A triangle has 2 sides, 112 and 120 cms. Point c in the triangle lies 65 cm from each vertex. Find the length of the third side ? Is this middle point a centroid or orthocenter ? Is there a formula we can use ? Thanks.

Another way:

from Wikipedia

so

130 = 2*(112)*(120)*c/√[(232+c)*(8+c)*(c-8)*(232-c)]

That's fourth order equation - will reduce to a quadratic quickly. Like Soroban would say - you solve for c, I'll wait in the car.