Altitudes

Geometry Level 2

True or False?

If any 2 altitudes of a triangle are known, then it is possible to calculate the range of the third altitude.

False True

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1 solution

Rishik Jain
Sep 10, 2016

True

Let the three altitudes be a , b , c a,b,c and their corresponding perpendicular sides be x , y , z x,y,z respectively. Then the area of triangle is given by: 1 2 a x = 1 2 b y = 1 2 c z \frac{1}{2}ax=\frac{1}{2}by=\frac{1}{2}cz

Then x = c z a , y = c z b x=\dfrac{cz}{a},y=\dfrac{cz}{b}

By triangle inequality, c z a + c z b > z c > a b a + b \dfrac{cz}{a}+\dfrac{cz}{b} \gt z \implies c \gt \dfrac{ab}{a+b}

Also, c z a + z > c z b c < a b a b \dfrac{cz}{a}+z \gt \dfrac{cz}{b} \implies c \lt \left|\dfrac{ab}{a-b}\right|

So, a b a + b < c < a b a b \boxed{\dfrac{ab}{a+b} \lt c \lt \left|\dfrac{ab}{a-b}\right|}

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