Question

If the ratio of sines of angles of a triangle is 1 : 1 : √2 then the ratio of square of the greatest side to sum of the squares of other two sides is
(a) 3 : 4
(b) 2 : 1
(c) 1 : 1
(d) 1 : 2

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{The ratio of sines of angles of a triangle is}\;\mathsf{1:1:\sqrt{2}}$

$\underline{\textbf{To find:}}$

$\textsf{The ratio of the square of the greatest side to}$

$\textsf{sum of the squares of other two sides}$

$\underline{\textbf{Solution:}}$

$\mathsf{Let\;\triangle\;ABC\;be\;the\;given\;triangle}$

$\mathsf{sinA:sinB:sinC=1:1:\sqrt{2}}$

$\mathsf{Using\;Sine\;formula,}$

$\mathsf{\dfrac{a}{sinA}=\dfrac{b}{sinB}=\dfrac{c}{sinC}=2R}$

$\mathsf{\dfrac{a}{1}=\dfrac{b}{1}=\dfrac{c}{\sqrt{2}}=2R}$

$\implies\mathsf{a=2R,\;b=2R,\;c=2\sqrt{2}R}$

$\textsf{Clearly, c is the greatest side}$

$\mathsf{Now,}$

$\mathsf{c^2:(a^2+b^2)}$

$\mathsf{=(2\sqrt{2}R)^2:((2R)^2+(2R)^2)}$

$\mathsf{=8R^2:(4R^2+4R^2)}$

$\mathsf{=8R^2:8R^2}$

$\mathsf{=1:1}$

$\underrline{\textbf{Answer:}}$

$\textsf{Option (c) is correct}$

Answer 2

Answer:

option: c

Step-by-step explanation:

given that the

square of the greatest side to sum of the squares of other two sides

by assuming√2 was the greatest side and other two sides were 1:1 as given by the question

we can say that

√2 square = sum of (1 +1)

2:2

=1:1

triangle has 3 sides