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?\\n
Answered by HeferH last updated on 26/Feb/23
$$\mathrm{18}\:? \\ $$
Commented byBaliramKumar 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 byBaliramKumar 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](Q188254.png)
$${p}\:=\:\mathrm{16} \\ $$ $${If}\:\:\:'{p}'\:\:\:{is}\:\:{even}\:\:\:\: \\ $$ $${No}.\:{of}\:\bigtriangleup\:=\:\left[\frac{{p}^{\mathrm{2}} }{\mathrm{48}}\right]\:=\:\left[\frac{\mathrm{16}×\mathrm{16}}{\mathrm{48}}\right]\:=\:\left[\mathrm{5}.\overset{−} {\mathrm{3}}\right]\:=\:\mathrm{\color{mathred}{5}} \\ $$ $${\color{mathbrown}{I}\color{mathbrown}{f}}\color{mathbrown}{\:}\color{mathbrown}{\:}\color{mathbrown}{'}{\color{mathbrown}{p}}\color{mathbrown}{'}\color{mathbrown}{\:}\color{mathbrown}{\:}{\color{mathbrown}{i}\color{mathbrown}{s}}\color{mathbrown}{\:}{\color{mathbrown}{o}\color{mathbrown}{d}\color{mathbrown}{d}} \\ $$ $${\color{mathbrown}{N}\color{mathbrown}{o}}\color{mathbrown}{.}\color{mathbrown}{\:}{\color{mathbrown}{o}\color{mathbrown}{f}}\color{mathbrown}{\:}\color{mathbrown}{\bigtriangleup}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\color{mathbrown}{\left[}\frac{\color{mathbrown}{\left(}{\color{mathbrown}{p}}\color{mathbrown}{+}\mathrm{\color{mathbrown}{3}}\color{mathbrown}{\right)}^{\mathrm{\color{mathbrown}{2}}} }{\mathrm{\color{mathbrown}{4}\color{mathbrown}{8}}}\color{mathbrown}{\right]} \\ $$ $$\:\color{mathblue}{\left[}{\color{mathblue}{x}}\color{mathblue}{\right]}\color{mathblue}{\:}\color{mathblue}{=}\color{mathblue}{\:}{\color{mathblue}{N}\color{mathblue}{e}\color{mathblue}{a}\color{mathblue}{r}\color{mathblue}{e}\color{mathblue}{s}\color{mathblue}{t}}\color{mathblue}{\:}{\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{t}\color{mathblue}{e}\color{mathblue}{g}\color{mathblue}{e}\color{mathblue}{r}}\color{mathblue}{\:}{\color{mathblue}{f}\color{mathblue}{u}\color{mathblue}{n}\color{mathblue}{c}\color{mathblue}{t}\color{mathblue}{i}\color{mathblue}{o}\color{mathblue}{n}} \\ $$ $$ \\ $$
Commented bymanxsol last updated on 27/Feb/23
$${deduction}\:{formula}? \\ $$