Question

The acute angle of the rhombus whose side is a mean proportional between its diagonals will be:

• A. ${15}^{\circ }$
• B. ${20}^{\circ }$
• C. ${30}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

Answer: (A)

${15}^{\circ }$

Let the diagonal are x,y in length S0

${\left(side\right)}^2=xyxy={\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{2}\right)}^{2}xy=\frac{x^2}{4}+\frac{y^2}{4}{x}^2+y^2-4xy=0\frac{x}{y}+\frac{y}{x}=4\frac{y}{x}+\frac{1}{\frac{y}{x}}=4let\frac{y}{x}=t{t}^2-4t+1=0t=\frac{4\pm \sqrt{16-4}}{2}\frac{y}{x}=\frac{4\pm \sqrt{12}}{2}\theta =\frac{y}{x}=2-\sqrt{3}\tan 15=\tan (45-30)=1-\frac{1}{\sqrt{3}}=2-\sqrt{3}\therefore \theta =15$