Question Number 188239 by BaliramKumar last updated on 26/Feb/23
$$ \\ $$The perimeter of a triangle is 16 units. How many triangles with integer sides can be made?
Answered by HeferH last updated on 26/Feb/23
$$\mathrm{18}\:? \\ $$
Commented by BaliramKumar last updated on 27/Feb/23
$${answer}\:{is}\:\mathrm{5} \\ $$
Answered by Frix last updated on 27/Feb/23
$$\mathrm{Let}\:{a}\leqslant{b}\leqslant{c} \\ $$$${a}\:\:\:\:\:{b}\:\:\:\:\:{c} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{7}\:\:\:\:\:\mathrm{7} \\ $$$$\mathrm{3}\:\:\:\:\:\mathrm{6}\:\:\:\:\:\mathrm{7} \\ $$$$\mathrm{4}\:\:\:\:\:\mathrm{5}\:\:\:\:\:\mathrm{7} \\ $$$$\mathrm{4}\:\:\:\:\:\mathrm{6}\:\:\:\:\:\mathrm{6} \\ $$$$\mathrm{5}\:\:\:\:\:\mathrm{5}\:\:\:\:\:\mathrm{6} \\ $$
Commented by BaliramKumar last updated on 27/Feb/23
$${nice} \\ $$
Answered by BaliramKumar last updated on 27/Feb/23
![p = 16 If ′p′ is even No. of △ = [(p^2 /(48))] = [((16×16)/(48))] = [5.3^− ] = 5 If ′p′ is odd No. of △ = [(((p+3)^2 )/(48))] [x] = Nearest integer function](https://www.tinkutara.com/question/Q188254.png)
$${p}\:=\:\mathrm{16} \\ $$$${If}\:\:\:'{p}'\:\:\:{is}\:\:{even}\:\:\:\: \\ $$$${No}.\:{of}\:\bigtriangleup\:=\:\left[\frac{{p}^{\mathrm{2}} }{\mathrm{48}}\right]\:=\:\left[\frac{\mathrm{16}×\mathrm{16}}{\mathrm{48}}\right]\:=\:\left[\mathrm{5}.\overset{−} {\mathrm{3}}\right]\:=\:\mathrm{5} \\ $$$${If}\:\:'{p}'\:\:{is}\:{odd} \\ $$$${No}.\:{of}\:\bigtriangleup\:=\:\left[\frac{\left({p}+\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{48}}\right] \\ $$$$\:\left[{x}\right]\:=\:{Nearest}\:{integer}\:{function} \\ $$$$ \\ $$
Commented by manxsol last updated on 27/Feb/23
$${deduction}\:{formula}? \\ $$