Question

The angles of a triangle are in the ratio $1:2:7$. Then the ratio of the greatest side of the triangle to the least side of the triangle is:

• A. $\sqrt{5}-1:\sqrt{5}+1$
• B. $\sqrt{5}+1:\sqrt{5}-1$
• C. $\sqrt{5}:\sqrt{5}+1$
• D. $\sqrt{5}-1:\sqrt{5}$

Answer 1

The correct option is B $\sqrt{5}+1:\sqrt{5}-1$

$A=x$

$B=2x$

$C=7x$

$A+B+C=180$

$x+2x+7x=180$

$x=18$

$B=36,C={126}^{\circ }$

$\frac{a}{\sin A}=\frac{c}{\sin C}$

$\frac{a}{\sin 18}=\frac{c}{\sin 126}$

$\frac{c}{a}=\frac{\sin (90+36)}{\sin 18}$

$=\frac{\cos 36}{\sin 18}$

$=\frac{\frac{\sqrt{5}+1}{4}}{\frac{\sqrt{5}-1}{4}}$

Thus the answer is $\frac{\sqrt{5}+1}{\sqrt{5}-1}$