which-rectangle-with-integer-length-side-have-numerically-the-same-area-and-perimeter-Find-them-all-Find-a-proof-that-convinces-that-you-have-found-them-all-what-about-right-angled-triang

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Question Number 170212 by otchereabdullai@gmail.com last updated on 18/May/22

$$\mathrm{which}\mathrm{rectangle}\mathrm{with}\mathrm{integer}\mathrm{length}$$$$\mathrm{side}\mathrm{have}\mathrm{numerically}\mathrm{the}\mathrm{same}\mathrm{area}$$$$\mathrm{and}\mathrm{perimeter}?\mathrm{Find}\mathrm{them}\mathrm{all}.\mathrm{Find}$$$$\mathrm{a}\mathrm{proof}\mathrm{that}\mathrm{convinces}\mathrm{that}\mathrm{you}\mathrm{have}$$$$\mathrm{found}\mathrm{them}\mathrm{all}.\mathrm{what}\mathrm{about}\mathrm{right}-$$$$\mathrm{angled}\mathrm{triangle}?\mathrm{how}\mathrm{many}\mathrm{solutions}?$$

Answered by aleks041103 last updated on 18/May/22

$$rectangle-a,b$$$$\Rightarrow ab=a+b$$$$ab-a-b+1=1$$$$a(b-1)-(b-1)=1$$$$(a-1)(b-1)=1$$$$a,b⩾0andareintegers$$$$\Rightarrow a-1\mid 1andb-1\mid 1$$$$butofallnaturalnumbers,only1\mid 1.$$$$\Rightarrow a-1=b-1=1\Rightarrow a=b=2$$$$\Rightarrow Squareofsidelength2istheonly$$$$soln.$$$$$$

Commented by Rasheed.Sindhi last updated on 18/May/22

$$\cap \mathrm{i}\subset \in !$$

Commented by otchereabdullai@gmail.com last updated on 18/May/22

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}\mathrm{am}\mathrm{much}\mathrm{grateful}$$$$\mathrm{for}\mathrm{your}\mathrm{time}!$$

Commented by floor(10²Eta[1]) last updated on 18/May/22

$$\mathrm{the}\mathrm{perimeter}\mathrm{is}2\mathrm{a}+2\mathrm{b}$$$$$$

Commented by floor(10²Eta[1]) last updated on 18/May/22

$$\mathrm{man}\mathrm{if}\mathrm{the}\mathrm{side}\mathrm{of}\mathrm{the}\mathrm{square}\mathrm{is}2\mathrm{the}\mathrm{perimeter}\mathrm{is}8$$

Commented by aleks041103 last updated on 19/May/22

$$Yes,{you}^{\prime }reright.Sorry!$$

Answered by aleks041103 last updated on 18/May/22

$$forrigthtriangles:$$$$want:xy=2(x+y+\sqrt{x^2+y^2})$$$$sincex,yareintegers,weneed\sqrt{x^2+y^2}to$$$$beaninteger.Thiscanhappenifwe$$$$haveapythagoriantriple.$$$$Allofthemaregeneratedby:$$$$x=u^2-w^2$$$$y=2uw$$$$\sqrt{x^2+y^2}=u^2+w^2$$$$\Rightarrow (u^2-w^2)2uw=2(u^2-w^2+2uw+u^2+w^2)$$$$\Rightarrow uw(u^2-w^2)=2u(u+w)$$$$\Rightarrow w(u-w)(u+w)=2(u+w)$$$$\Rightarrow w(u-w)=2$$$$Sincex,y\in \mathbb{N},u,w\in \mathbb{N}$$$$\Rightarrow w\mid 2andu-w\mid 2$$$$\Rightarrow w=1,2andu-w=2,1$$$$\Rightarrow (u,w)\in \{(3,1),(3,2)\}$$$$\Rightarrow (x,y)\in \{(8,6),(5,12)\}$$$$andreally$$$$\bullet x=8,y=6,z=\sqrt{x^2+y^2}=10$$$$\frac{1}{2}xy=24=6+8+10✓$$$$\bullet x=5,y=12,z=13$$$$\frac{1}{2}xy=30=5+12+13✓$$

Commented by Rasheed.Sindhi last updated on 18/May/22

$$\mathcal{N}ice!$$

Commented by otchereabdullai@gmail.com last updated on 18/May/22

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}\mathrm{am}\mathrm{much}\mathrm{grateful}$$$$\mathrm{for}\mathrm{your}\mathrm{time}!$$

Answered by floor(10²Eta[1]) last updated on 18/May/22

$$\mathrm{ab}=2\mathrm{a}+2\mathrm{b}$$$$2\mathrm{a}-\mathrm{ab}+2\mathrm{b}=0$$$$\mathrm{a}(2-\mathrm{b})-2(2-\mathrm{b})=-4$$$$(2-\mathrm{a})(2-\mathrm{b})=4$$$$2-\mathrm{a}=-1$$$$2-\mathrm{b}=-4$$$$\mathrm{a}=3,\mathrm{b}=6$$$$2-\mathrm{a}=-2$$$$2-\mathrm{b}=-2$$$$\mathrm{a}=\mathrm{b}=4$$$$\mathrm{all}\mathrm{sol}:\{(4,4),(3,6),(6,3)\}$$

Commented by Rasheed.Sindhi last updated on 18/May/22

$$\vee .\cap \mathrm{i}\subset \in !$$

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