Three equations with square variables

Let

a^2+b^2+ab=9

b^2+c^2+bc=16

a^2+c^2+ac=25

What is the value of ab + bc +ac?

Note by Sinuhé Ancelmo
5 years, 10 months ago

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3 votes

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Take a point OO from which three line segments OA,OB,OCOA, OB, OC, lying on the same plane, emanate such that their respective lengths are a,b,ca, b, c and the angle between any two of them is 120 degrees. Use Cosine rule and the mentioned equations to find AB,BC,CAAB, BC, CA which will have values 3,4 and 5 units. Now this triangle ABCABC is right angled( by the converse of Pythagoras' Theorem), so it's area will be 6 sq. units. Also, the area of the three triangles AOB,BOC,COAAOB, BOC, COA can be found out by the sine rule (individually), and now equate it with the earlier found area. The answer comes out to be 838 \sqrt{3}. [ the calculations are to be done by you].

Shourya Pandey - 5 years, 10 months ago

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Well, your idea is interesting, but are you sure about the value of ab + bc +ac? shouldn' it be 8*sqrt(3)? Could you chek it again, plz??

Sinuhé Ancelmo - 5 years, 10 months ago

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Edited.

Shourya Pandey - 5 years, 10 months ago

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@Shourya Pandey You can realize that - 8 * sqrt (3) is also a valid answer for ab + bc + ac, right?

Sinuhé Ancelmo - 5 years, 10 months ago

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@Shourya Pandey Thanks! Already read your edition, friend!

Sinuhé Ancelmo - 5 years, 10 months ago

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Could you show any algebraic, only algebraic solution, friend?

Sinuhé Ancelmo - 5 years, 10 months ago

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Is it possible to solve this algebraically?

Ajit Athle - 5 years, 10 months ago

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The solution could be algebraical or geometrical, Thanks in advance for help!

Sinuhé Ancelmo - 5 years, 10 months ago

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what will be the algebric solution Help me

Anit Singh - 4 years, 2 months ago

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The point "O" is known as "Fermat" or Torricelli Point of the triangle.

Ajit Athle - 11 months, 3 weeks ago

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