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In triangle PQR, p,q,r in G.P. and p^2,q^2,r^2 \ are\ in\ A.P. then which of the following is true about triangle PQR

Option: 1

Triangle\ PQR \ is\ a\ scalene\ triangle


Option: 2

Triangle\ PQR \ is\ a\ isosceles \ triangle


Option: 3

Triangle\ PQR \ is\ a\ equilateral\ triangle


Option: 4

Can't say anything 


Answers (1)

best_answer

Basic relation b/w sides and angle of triangle and Sine Rule -

Basic relation b/w sides and angle of triangle and Sine Rule

 

In this section, Properties and solution of Triangle. We will be using some standard symbols.

 In ΔABC, the angles are denoted by capital letters A, B and C and the length of the sides opposite to these angles are denoted by small letters a, b and c respectively.

 

\mathrm{\begin{array}{l}{ \angle B A C=A} \\ { \angle A B C=B} \\ { \angle B C A=C}\end{array}}

 

Sides of the ΔABC

AB = c, AC = b, and  BC = a

Collectively, these relationships are called the Law of Sines

 

\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}

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p,q,r \ in\ G.P. \rightarrow q^2=pr\\ p^2,q^2,r^2 \ are\ in\ A.P.\rightarrow 2q^2=p^2+r^2\\ 2pr=p^2+r^2\\ (p-r)^2=0\\ p=r\\ \because q^2=pr\\ p=q=r\\ Triangle\ PQR \ is\ a\ equilateral\ trianglep,q,r \ in\ G.P. \rightarrow q^2=pr\\ p^2,q^2,r^2 \ are\ in\ A.P.\rightarrow 2q^2=p^2+r^2\\ 2pr=p^2+r^2\\ (p-r)^2=0\\ p=r\\ \because q^2=pr\\ p=q=r\\ Triangle\ PQR \ is\ a\ equilateral\ triangle

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