## inkyvoyd 2 years ago Simple geometry

1. inkyvoyd

|dw:1352774312422:dw| how does one express

2. inkyvoyd

e in terms of a, b, c, and d?

3. AccessDenied

By Pythagorean Theorem: $$b^2 + e^2 = a^2$$, $$d^2 + e^2 = c^2$$ Solving for $$e^2$$ in both cases: $$e^2 = a^2 - b^2$$ $$e^2 = c^2 - d^2$$ I guess one way we could do so is by simply adding the equations together... $$2e^2 = e^2 + e^2$$ $$2e^2 = (a^2 - b^2) + (c^2 - d^2)$$ $$\displaystyle e^2 = \frac{a^2 + c^2 - (b^2 + d^2)}{2}$$ $$\displaystyle e = \sqrt{\frac{a^2 + c^2 - (b^2 + d^2)}{2}}$$

4. inkyvoyd

Question, if a and c were known, but only b+d was known, how would one solve htis problem?

5. inkyvoyd

I mean, say that you had 3 sides of a triangle, and drew a given altitude, what would be the two lengths of the resulting split side?

6. AccessDenied

Hm, I think we would have to find e in terms of a and c using angles (found by law of cosines). I don't think it would be solvable with that particular equation only since you'd only have two equations for three unknowns there: e, b, and d...

7. inkyvoyd

But it would be solvable - draw any triangle right now, and you can measure the sides - now draw an altitude to any single side - you have just created the problem.

8. inkyvoyd

Apparently I can do this with heron's theorem and the pythagorean theorem, but I was wondering if there are a few lines or simple algebraic manipulations I could make to get it done easily. http://en.wikipedia.org/wiki/Altitude_(triangle) wher it says "altitude in terms of the sides"