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If in Delta ABC, 8R^(2) = a^(2) + b^(2) ...

If in `Delta ABC, 8R^(2) = a^(2) + b^(2) + c^(2)`, then the triangle ABC is

A

right angled

B

isosceles

C

equilateral

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
We have
`8R^(2) = a^(2) + b^(2) + c^(2)`
`4 R^(2) (sin^(2) A + sin^(2) B + sin^(2)C)`
or `sin^(2) A + sin^(2)B + sin^(2)C = 2`
or `1 - cos^(2) A + 1 - cos^(2) B + sin^(2) C = 2`
or `(cos^(2) A - sin^(2) C) + cos^(2) B = 0`
or `cos(A + C) cos (A -C) + cos^(2) B = 0`
or `-cosB [cos (A -C) - cosB] = 0`
or `-cos B [cos (A -C) + cos (A +C)] = 0`
or `-2 cos A cos B cos C = 0`
or `cos A = 0 " or " cos B " or " cos C = 0`
or `A = (pi)/(2) " or " B = (pi)/(2) " or " C = (pi)/(2)`
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CENGAGE-PROPERTIES AND SOLUTIONS OF TRIANGLE-Exercise (Single)
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  2. With usual notations, in triangle ABC, a cos (B - C) + b cos (C - A) +...

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  3. If in Delta ABC, 8R^(2) = a^(2) + b^(2) + c^(2), then the triangle ABC...

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  4. Let ABC be a triangle with /A=45^0dot Let P be a point on side BC with...

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  5. In any triangle ABC, (a^(2) + b^(2) + c^(2))/(R^(2)) has the maximum v...

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  6. In triangle ABC, R (b + c) = a sqrt(bc), where R is the circumradius o...

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  7. In DeltaABC, " if " b^(2) + c^(2) = 2a^(2), then value of (cotA)/(cot ...

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  8. If sin theta and -cos theta are the roots of the equation ax^(2) - bx ...

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  9. If D is the mid-point of the side B C of triangle A B C and A D is per...

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  10. In a triangle ABC, if cotA :cotB :cotC = 30: 19 : 6 then the sides a, ...

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  11. In Delta ABC, P is an interior point such that angle PAB = 10^(@), ang...

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  12. In DeltaABC, if AB = c is fixed, and cos A + cosB + 2 cos C = 2 then t...

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  13. If in A B C ,A=pi/7,B=(2pi)/7,C=(4pi)/7 then a^2+b^2+c^2 must be R^2 ...

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  14. In Delta ABC, "cot"(A)/(2) + "cot" (B)/(2) + "cot" (C)/(2) is equal to

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  15. In Delta ABC, (cot. (A)/(2) + cot. (B)/(2)) (a sin.^(2) (B)/(2) + b si...

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  16. In a right-angled isosceles triangle, the ratio of the circumradius an...

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  17. In the given figure, what is the radius of the inscribed semicircle ha...

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  18. In DeltaABC, A = (2pi)/(3), b -c = 3 sqrt3 cm and " area of " Delta AB...

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  19. In triangle ABC, let angle C = pi//2. If r is the inradius and R is ci...

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  20. In the given figure, AB is the diameter of the circle, centered at O. ...

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