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inkyvoyd
Simple geometry
|dw:1352774312422:dw| how does one express
e in terms of a, b, c, and d?
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... \(2e^2 = e^2 + e^2\) \(2e^2 = (a^2 - b^2) + (c^2 - d^2)\) \(\displaystyle e^2 = \frac{a^2 + c^2 - (b^2 + d^2)}{2}\) \(\displaystyle e = \sqrt{\frac{a^2 + c^2 - (b^2 + d^2)}{2}}\)
Question, if a and c were known, but only b+d was known, how would one solve htis problem?
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?
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...
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.
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"