Consider-point-A-inside-a-triangle-with-sides-3-4-and-5-if-d-is-the-sum-of-the-distances-of-this-point-from-the-sides-what-is-the-smallest-value-of-d-

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Question Number 204657 by es last updated on 24/Feb/24

Consider point A inside a triangle with sides 3,4 and 5. if d is the sum of the distances of this point from the sides.what is the smallest value of d?

$${Consider}\:{point}\:{A}\:{inside}\:{a}\:{triangle} \\ $$$${with}\:{sides}\:\mathrm{3},\mathrm{4}\:{and}\:\mathrm{5}.\:{if}\:{d}\:\:{is}\:{the}\:{sum} \\ $$$$\:{of}\:{the}\:{distances}\:\:{of}\:{this}\:{point}\:{from} \\ $$$${the}\:{sides}.{what}\:{is}\:{the}\:{smallest} \\ $$$${value}\:{of}\:{d}? \\ $$$$ \\ $$

Answered by mr W last updated on 24/Feb/24

d_(min) =((3×4)/5)=2.4 d_(max) =((3×4)/3)=4

$${d}_{{min}} =\frac{\mathrm{3}×\mathrm{4}}{\mathrm{5}}=\mathrm{2}.\mathrm{4} \\ $$$${d}_{{max}} =\frac{\mathrm{3}×\mathrm{4}}{\mathrm{3}}=\mathrm{4} \\ $$

Commented by mr W last updated on 25/Feb/24

generally: d_(min) =((2Δ)/(max(a,b,c))) d_(max) =((2Δ)/(min(a,b,c))) with Δ=area of triangle

$${generally}: \\ $$$${d}_{{min}} =\frac{\mathrm{2}\Delta}{{max}\left({a},{b},{c}\right)} \\ $$$${d}_{{max}} =\frac{\mathrm{2}\Delta}{{min}\left({a},{b},{c}\right)} \\ $$$${with}\:\Delta={area}\:{of}\:{triangle} \\ $$

Commented by mr W last updated on 26/Feb/24

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