# Solving for height in back to back pythagorus triagles

### Solving for height in back to back pythagorus triagles

Hi I am have a real life problem, but somehow cannot get the 'b' side of the equation and think I am making the wrong assumption for the "c's"

I have attached the model of my problem, and I know the following:

The a's: Hin, Hout
The b's: (b+Vin), (b+Vout), where I put in negative for values under the b datum.
The c's: I know the total length, Ctot, between Cin and Cout, and made the assumptions Ctot=Cin+Cout and Ctot^2=Cin^2+Cout^2

And knowing the Pythagoras theorem, and rearranging to get a quadratic equation I can find b. But I think I am making the wrong assumption for the formula for the Ctot, with respect to Cin and Cout. I need help on this part of the assumption for my formula.

This is what I got:
Combining two Pythagoras equations, and making them equal 0
($$Hin^{2}$$+$$(B+Vin)^{2}$$-$$Cin^{2}$$)+($$Hout^{2}$$+$$(B+Vout)^{2}$$-$$Cout^{2}$$)=0

Expanding I get the below, and think I made a mistake for the last Ctot part of the expression, to get Ctot in terms of Cin and Cout.
2*$$b^{2}$$+b*2*(Vin+Vout)+$$Vin^{2}$$+$$Vout^{2}$$+$$Hin^{2}$$+$$Hout^{2}$$-$$Ctot^{2}$$

When solving using the quatdratic equation, I get a b value. But, when I take my b and substitute for each triangle, and sum to verify the Ctot. I don't get the same Ctot that is known.

Please help, as I need another person's set of eyes to possible see what I am doing wrong.

Regards and Thank-you,
Miguel
Attachments
Loop_InOut.pdf
A buffer loop system and with material
Guest

### Re: Solving for height in back to back pythagorus triagles

I got a little confused by the variables, is the below correct?

The horizontal legs are known.
The vertical legs are unknown, but the difference between them, ($$v_{in} + v_{out}$$), is known.
The hypotenuses are unknown, but their sum is known.

If so, the problem should be solvable. You have 3 unknowns (b and both C's) in 3 equations (2 Pythagorean relations and the sum of the C's). The usual method is to use elimination and substitution to get a quadratic in one of the variables. I'll type it up if you like. I just want to first make sure I'm solving the right problem since all the named constants make it messy.

phw

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### Re: Solving for height in back to back pythagorus triagles

Thank-you for the help. Yes you are correct, in the assumptions. I know my total length of the 2 hypotenuse, but not the individual Cin and Cout, and the a's. I am trying to find b, so that it satisfies both the b's of the two triangles, knowing the differences from b zero datum that I have. Those are the Vin and Vout values. I had drawn out the model, but forget to put the "b" to be the total length, below the b-0 datum. I got a quadratic equation, from what I posted earlier, but its not correct. I worked it out a few times to make sure, and how to handle the Cin and Cout, I got stumped.

I tried to make it easy to understand my problem, but to make the algebra easier I put in H for horizontal of the "a" values, V for the vertical of the "b" values.

Thank-you again for the help.
Guest

### Re: Solving for height in back to back pythagorus triagles

Phw,

If you could help with the writing of the quadratic equation that would be great. I know its a little messy, but I tried to make it as general as possible, because the real life situation of the triangles can be changed. And, if that were to happen I would like to put in the new knowns, and easily solve for "b".
Guest