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Standard X

Mathematics

Relation between Inradius and Perimeter of Triangle

The figure sh...

Question

The figure shows an equilateral triangle ABC inscribed in a circle.D is the midpoint of side AC and the equilateral triangle DEF is inscribed in the circular segment AC. If AB =12 , find [DE] where [x] represents the Greatest Integer value of x.

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Solution

In $Δ O D E$
$O E = R = 4 \sqrt{3}$
DE = x
$O D = 2 \sqrt{3} = r$
$∠ O D E = 150^{0}$ so cosine Rule
$c o s 150^{0} = \frac{(2 \sqrt{3})^{2} + x^{2} - (4 \sqrt{3})^{2}}{2 \times 2 \sqrt{3} . x} = \frac{- \sqrt{3}}{2}$
$\Rightarrow x^{2} - 36 = - 6 x$
$\Rightarrow x^{2} + 6 x - 36 = 0 \Rightarrow x = \frac{- 6 \sqrt{36 + 4.36}}{2}$
so , [x] = 3 $x = 3 (\sqrt{5} - 1)$

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The figure shows an equilateral triangle ABC inscribed in a circle.D is the midpoint of side AC and the equilateral triangle DEF is inscribed in the circular segment AC. If AB =12 , find [DE] where [x] represents the Greatest Integer value of x.

In ΔODEOE=R=4√3DE = xOD=2√3=r∠ODE=1500 so cosine Rulecos1500=(2√3)2+x2−(4√3)22×2√3.x=−√32⇒x2−36=−6x⇒x2+6x−36=0⇒x=−6√36+4.362so , [x] = 3 x=3(√5−1)