lf $r : R : r_{1} = 2 : 5 : 12$ , then $A =$

45^{o}

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60^{o}

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30^{o}

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90^{o}

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Solution

The correct option is C $90^{o}$
$r = 2 k, r_{1} = 12 k, R = 5 k$
$r_{1} = s tan (A / 2) r = (s - a) tan (A / 2)$

$\frac{r_{1}}{r} = \frac{s}{s - a}$

$\frac{r}{r_{1}} = 1 - \frac{a}{s}$

$\frac{1}{6} = 1 - \frac{a}{s}$

$\frac{a}{s} = \frac{5}{6}$

$R = \frac{a}{2 sin A}$

$r_{1} = s tan A / 2$

$\frac{5}{12} = \frac{a}{2 s sin A tan A / 2}$

$sin A tan A / 2 = 1$
$2 {sin}^{2} (A / 2) = 1$
$sin A / 2 = \frac{1}{\sqrt{2}}$
$A / 2 = 45$
$A = 90^{0}$

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