In a triangle a 2- b 2- c 2- ca - ab -3- then triangle isA- Right angledB- None of theseC- EquilateralD- Right angled and isosceles

The correct option is A Right angledGiven, a^2+b^2+c^2=ca+ab√(3) a^2+b^2+c^2 ca ab√(3)=0 ⇒ ( a√(3)/2 b )^2+ ( a/2 c )^2 = 0 ⇒ a=2b/√(3) and a=c/2 We can ...

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Standard XII

Mathematics

Sine Rule

In a triangle...

Question

In a triangle $a^{2} + b^{2} + c^{2} = c a + a b \sqrt{3}$ , then triangle is

Equilateral

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Right angled and isosceles

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Right angled

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None of these

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Solution

The correct option is C Right angled
Given, $a^{2} + b^{2} + c^{2} = c a + a b \sqrt{3} a^{2} + b^{2} + c^{2} - c a - a b \sqrt{3} = 0 \Rightarrow {(\frac{a \sqrt{3}}{2} - b)}^{2} + {(\frac{a}{2} - c)}^{2} = 0 \Rightarrow a = \frac{2 b}{\sqrt{3}} a n d a = \frac{c}{2} W e c a n s e e, a^{2} + b^{2} = c^{2} s i n B = \frac{b}{a} = \frac{\sqrt{3}}{2} \neq 1 \Rightarrow N o t i s o s c e l e s ∴ G i v e n t r i a n g l e i s a R i g h t a n g l e d t r i a n g l e w h i c h i s n o t i s o s c e l e s$ .
Option C is correct.

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In a triangle a2+b2+c2=ca+ab√3, then triangle is

The correct option is C Right angledGiven, a2+b2+c2=ca+ab√3a2+b2+c2−ca−ab√3=0⇒(a√32−b)2+(a2−c)2=0⇒a=2b√3 and a=c2We can see, a2+b2=c2sinB=ba=√32≠1⇒Not isosceles∴Given triangle is a Right angled triangle which is not isosceles. Option C is correct.

In a triangle $a^{2} + b^{2} + c^{2} = c a + a b \sqrt{3}$ , then triangle is