Trisected Line Segment of a Scalene Triangle. What is the length of the trisected angle given all three sides of the scalene?

Question

I keep on getting away from the answer by about the same number, either upper or lower. Please help me find CD. The problem is found in the link below. It is slightly different from my problem, but I have already found the 3 sides as well, so I would like to know how to solve for line segment CD of the scalene given the 3 sides. I have tried Trig, to using Pythagoras's theorem in mix with Trig, and can't seem to reach the right answer. Please help.

 

http://postimg.org/image/3rb124ugr/

Answer 1

First, remember that the Pythagorean Theorem is only applicable for right triangles. So, if you have no right triangles, you can't use the theorem. The same goes for applying SOHCAHTOA, since non-right triangles don't have hypotenuses.

 

 

 

For the image you posted, first use the Law of Cosines on ΔABD to find ∠ABD:

 

c² = a² + b² - 2ab(cos(∠C))

71.5² = 81.2² + 70.3² - 2(81.2)(70.3)cos(∠ABD)

5112.25 = 11535.53 - 11416.72(cos(∠ABD))

0.56262 = cos(∠ABD)

∠ABD ≈ 55.76 degrees

 

Now, use the Law of Cosines on ΔBCD to find CD. Remember that, since AB is trisected, CB is 81.2/3 = 27.06667.

 

(CD)² = 27.06667² + 70.3² - 2(27.06667)(70.3)cos(55.76 degrees)

(CD)² = 5674.69 - 3805.57(0.56262)

(CD)² = 3533.6

CD ≈  59.44

Answer 2

Using Law of Cosines, we can set up two equations...

 

1. BD² = AD² + AB² - 2 AD AB cosA

 

2. CD² = AD² + AC² - 2 AD AC cosA

 

Solving equation 1. for 2 AD cosA gives us...

 

2 AD cosA = ( AD² + AB² - BD² ) / AB

 

... and solving equation 2. for 2 AD cosA...

 

2 AD cosA = ( AD² + AC² - CD² ) / AC

 

Now we can say that...

 

( AD² + AB² - BD² ) / AB  =  ( AD² + AC² - CD² ) / AC

 

... because both sides of the equation are equal to 2 AD cosA. Solving for CD should leave us with...

 

CD = √( AD + AC² - ( AC ⁄ AB ) ( AD² + AB² - BD² ) )

 

Because AB is trisected, we know that...

 

AC = ( 2 AB ) / 3

 

So we can substitute ( 2 AB ) / 3 for all instances of AC, leaving us with...

 

CD = √( AD + (4 AB² / 9) - ( 2 / 3 ) ( AD² + AB² - BD² ) )

 

Now we have CD in terms the three sides that have known lengths, which should be different on your particular problem.