True or False?
If any 2 altitudes of a triangle are known, then it is possible to calculate the range of the third altitude.
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True
Let the three altitudes be a , b , c and their corresponding perpendicular sides be x , y , z respectively. Then the area of triangle is given by: 2 1 a x = 2 1 b y = 2 1 c z
Then x = a c z , y = b c z
By triangle inequality, a c z + b c z > z ⟹ c > a + b a b
Also, a c z + z > b c z ⟹ c < ∣ ∣ ∣ ∣ a − b a b ∣ ∣ ∣ ∣
So, a + b a b < c < ∣ ∣ ∣ ∣ a − b a b ∣ ∣ ∣ ∣