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Question Number 35872 by Rasheed.Sindhi last updated on 25/May/18

Commented by Tinkutara last updated on 25/May/18

Yes this was also my doubt!

Answered by Rasheed.Sindhi last updated on 26/May/18

$$\mathrm{Note}:\mathrm{Capital}\mathrm{C}\mathrm{and}\mathrm{small}\mathrm{c}\mathrm{has}\mathrm{been}$$$$\mathrm{used}\mathrm{for}\mathrm{same}\mathrm{quantity}(\mathrm{the}\mathrm{greatest}$$$$\mathrm{side}\mathrm{of}\mathrm{the}\mathrm{triangle}\left(\mathrm{fix}\right))\mathrm{in}\mathrm{the}\mathrm{question}.$$$$\mathrm{Which}\mathrm{create}\mathrm{confusion}.$$$$\mathrm{Given}:$$$${}^{\bullet }\mathrm{For}\mathrm{the}\mathrm{triangle},\mathrm{whose}\mathrm{sides}\mathrm{are}a,b,c$$$$\left(\mathrm{i}\right)\mathrm{c}\in {\mathbb{O}}^{+}\mathrm{and}\mathrm{fixed}.\left(\mathrm{ii}\right)a⩽b⩽c$$$$\left(\mathrm{iii}\right)b=a\ne c\mid b=c\ne a\mid a=b=c$$$$$$$$\mathrm{The}\mathrm{triangle}\mathrm{may}\mathrm{be}\mathrm{of}\mathrm{following}\mathrm{type}:$$$$(a,a,c)\mathrm{or}(a,c,c)\mathrm{or}(c,c,c)$$$$$$$$(a,a,c):a<\mathrm{c}\wedge 2a>c$$$$a⩽\mathrm{c}-1\wedge 2a⩾c+1\Rightarrow a⩾\frac{c+1}{2}$$$$\frac{c+1}{2}⩽a⩽c-1$$$$a\mathrm{takes}\mathrm{value}\mathrm{from}\frac{c+1}{2}\mathrm{to}c-1\mathrm{inclusive}$$$$\therefore a\mathrm{takes}\mathrm{values}(c-1)-\left(\frac{c+1}{2}\right)+1=\frac{\mathrm{c}-1}{2}$$$$\therefore \mathrm{Number}\mathrm{of}(a,a,c)\mathrm{type}\mathrm{triangles}=\frac{\mathrm{c}-1}{2}$$$$$$$$(a,c,c):a<c\wedge a<2c$$$$\Rightarrow a<c\Rightarrow a⩽c-1$$$$a\mathrm{is}\mathrm{upto}c-1\Rightarrow c-1\mathrm{triangles}$$$$$$$$(c,c,c):c\mathrm{is}\mathrm{fixed}\left(\mathrm{only}1\mathrm{value}\right)$$$$1\mathrm{triangle}$$$$\mathrm{Total}\mathrm{triangles}=\left(\frac{c-1}{2}\right)+(c-1)+1$$$$=\frac{3c-1}{2}$$

Commented by Tinkutara last updated on 26/May/18

Thanks Sir!

Commented by Rasheed.Sindhi last updated on 26/May/18

$$\mathrm{Could}\mathrm{you}\mathrm{tell}\mathrm{me}\mathrm{the}\mathrm{name}\mathrm{and}\mathrm{author}$$$$\mathrm{of}\mathrm{the}\mathrm{book},\mathrm{which}\mathrm{contains}\mathrm{so}\mathrm{much}$$$$\mathrm{interesting}\mathrm{questions}?\mathrm{Is}\mathrm{there}\mathrm{its}\mathrm{soft}$$$$\mathrm{copy}\mathrm{on}\mathrm{the}\mathrm{net}?$$

Commented by Tinkutara last updated on 26/May/18

It is just a coaching material and not any standard book �� And not all the questions are interesting, being some too easy while some too hard ��

Commented by Rasheed.Sindhi last updated on 26/May/18

�� Anyway sir, we received some interesting questions of it through you!

Commented by Rasheed.Sindhi last updated on 26/May/18

You mentioned about the question: "Yes this was also my doubt!" What was that doubt?

Commented by Tinkutara last updated on 26/May/18

$$Thequestion.$$

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