The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side.
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Tags: geometry proposed, geometry
Goutham
30.10.2010 20:22
http://en.wikipedia.org/wiki/Viviani's_theorem
DrYouKnowWho
01.07.2020 06:25
$5\sqrt{3}-4$
GammaZero
23.12.2020 05:31
By Vivani's theorem we know that the distance from a point inside an equilateral triangle to each of the $3$ sides is going to be the same as the altitude of the triangle. Thus let us call the distance from $P$ to the third side as $\gamma.$ Then
$$1+3+\gamma=\sqrt{10^2-5^2}$$$$\gamma=\boxed{5\sqrt{3}-4}$$
OlympusHero
22.10.2021 17:13
From Viviani's Theorem, the sum of these three distances is equal to the altitude, which is $5\sqrt3$. The distance from $P$ to the third side is hence $5\sqrt3-1-3=\boxed{5\sqrt3-4}$.
e61442289
22.10.2021 20:12
WakeUp wrote: The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side. $5\sqrt{3}-4$ by Viviani's theorem