∵Sumofanglesoftriangle=180∘
⇒α+β=90∘
⇒α<90∘,β<90∘
sin(α)=ABAC
⇒AC=ABsinα
sinαϵ(0,1)
∴1sinα>1
⇒AC>AB−(1)
sinβ=BCAC
⇒AC=BCsinβ
sinβ∈(0,1)
⇒1sinβ>1
⇒AC>BC−(2)
From(1)\& (2),AC is the longest side.
Show that in a right angled triangle, the hypotenuse is the longest side.