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?

#Algebra #HelpMe! #MathProblem #Math

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Comments

Take a point $O$ from which three line segments $OA, OB, OC$ , lying on the same plane, emanate such that their respective lengths are $a, b, c$ and the angle between any two of them is 120 degrees. Use Cosine rule and the mentioned equations to find $AB, BC, CA$ which will have values 3,4 and 5 units. Now this triangle $ABC$ 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 $AOB, BOC, COA$ can be found out by the sine rule (individually), and now equate it with the earlier found area. The answer comes out to be $8 \sqrt{3}$ . [ the calculations are to be done by you].

Shourya Pandey - 7 years, 6 months ago

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??