Can Vector Addition Solve Non-Right Triangle Proofs?

[byronsakic]

Hello,
Here is the question:

Here are 2 different ways i tried to approach the question:
In the first way, i tried using pythagoreom theorem by making a right angle, but after i did the question i realized since it did not say it was a right triangle, ic annot assume there is a 90 degrees angle and i cannot use the pythagoras theorem.

Second way i tried using vector addition. However you can see at the end i was stuck.

though i believe vector addition is the way to solve this problem, however we had never done any problems involving the squared of a vector unless it was using the pythagoras theorem,
any suggestions in what i did wrong or what i could do?

Sorry for my messy writing hehe.

thanks
byron

 

[Tide]

Byron,

Try this. Drop a perpendicular from A to the line segment BC. Call its length h and the distance between the intersection point and the midpoint x. Now you can use Pythagoras to solve for x and h from which you can calculate AD.

 

[Hurkyl]

I know why you can't do it with vector addition: nowhere in your work did you make use of the fact that D is the midpoint of the line segment BC.

(In fact, you didn't even made use of the fact that D is on the line through B and C!)

 

[byronsakic]

didnt i do it in the first try? but if you drop a perpendicular from A to line segment BC then how would u prove that it is a right angle? but i think i did that in first try..

though i am going to try coordinate proofs right now :D i think that is the solution

 

[Hurkyl]

I only looked at your vector addition approach. I agree that ought to be the easiest solution. (unless you're particularly adept with synthetic geometry)

 

[byronsakic]

i got the answer by using coordinate proofs and making LS = RS

thanks for all your help :D