A $$12$$ cm wire is given a shape of a right angled triangle A B C having sides $$3$$ cm, $$4$$ cm and $$5$$ cm as shown in the figure. The resistance between two ends (A$$ B, BC, $$CA) of the respective sides are measured one by one by a $$multi-meter$$. The resistances will be in the ratio

Question

Infinity Learn · Accepted Answer

Let ρ and A be resistivity and area of $$cross-section$$ of the wire respectively. The wire is bent in the form of right triangle as shown in adjacent figure. Resistance of side AB is $$R1=$$ρ$$3A$$  Resistance of side BC is $$R2=$$ρ$$4A$$  Resistance of side AC is $$R3=$$ρ$$5A$$ ∴ The resistance between the ends A and B is  $$RAB=R1R2+R3R1+R2+R3=3$$ρ$$A4$$ρ$$A+5$$ρ$$A3$$ρ$$A+4$$ρ$$A+5$$ρ$$A=2712$$ρA  The resistance between the ends B and C is  $$RBC=R2R1+R3R2+R1+R3=4$$ρ$$A3$$ρ$$A+5$$ρ$$A4$$ρ$$A+3$$ρ$$A+5$$ρ$$A=3212$$ρA  The resistance between the ends A and C is  $$RAC=R3R1+R2R3+R1+R2$$ $$=5$$ρ$$A3$$ρ$$A+4$$ρ$$A5$$ρ$$A+3$$ρ$$A+4$$ρ$$A=3512$$ρA ∴  RAB:RBC:$$RAC=2712$$:$$3212$$:$$3512=27$$:$$32$$:$$35$$

A 12 cm wire is given a shape of a right angled triangle A B C having sides 3 cm, 4 cm and 5 cm as shown in the figure. The resistance between two ends (A B, BC, CA) of the respective sides are measured one by one by a multi-meter. The resistances will be in the ratio

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