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Question Number 44142 by Tawa1 last updated on 22/Sep/18

$$\mathrm{Find}\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{integral}\mathrm{sided}\mathrm{triangle}\mathrm{whose}\mathrm{largest}\mathrm{side}$$$$\mathrm{is}30.$$$$\mathrm{please}\mathrm{explain}\mathrm{with}\mathrm{workings}$$

Answered by MrW3 last updated on 22/Sep/18

$${let}^{\prime }ssaythelargestsideiscwith$$$$c=30$$$$a+b>c,a⩽b<30$$$$withb=29:a=2,3,...,29\Rightarrow 28triangles$$$$withb=28:a=3,4...,28\Rightarrow 26triangles$$$$withb=27:a=4,5,...,27\Rightarrow 24triangles$$$$......$$$$withb=16:a=15,16\Rightarrow 2triangles$$$$totalnumberofpossibletriangles:$$$$2+4+...+24+26+28=2(1+2+...+14)$$$$=2\times \frac{14\times (1+14)}{2}=14\times 15=210$$

Commented by Tawa1 last updated on 22/Sep/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{really}\mathrm{appreciate}$$

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