Q71 / 160 · AP EAMCET 2007 · Mathematics

If two angles of $\triangle A B C$ are $45^{\circ}$ and $60^{\circ}$, then the ratio of the smallest and the greatest sides are

  • A
    $(\sqrt{3}-1): 1$
  • B
    $\sqrt{3}: \sqrt{2}$
  • Correct answer
    C
    $1: \sqrt{3}$
  • D
    $\sqrt{3}: 1$

Solution

Let $A=45^{\circ}$ and $B=60^{\circ}$ $\therefore \quad C=75^{\circ}$ Let smallest and greatest sides are $a$ and $c$. $\therefore \quad \begin{aligned} a: c & =\sin 45^{\circ}: \sin 75^{\circ} \\ & =\frac{1}{\sqrt{2}}: \frac{\sqrt{3}+1}{2 \sqrt{2}} \\ & =1: \frac{\sqrt{3}+1}{2} \\ & =2: \sqrt{3}+1 \\ & =1: \sqrt{3}\end{aligned}$

Hence, the correct answer is option: 'C'.

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