Let A B C be the triangle with A B = 1 , A C = 3 and B A C = 2 . If a circle of radius r 0 touches the sides A B , A C and also touches internally the circumcircle of the triangle A B C , then the value of r is _______.

Question

Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]

Smart Achievers · Accepted Answer

(0.84)

$We have A B = 1, A C = 3 and ∠ B A C = \frac{π}{2}$

$r > 0$

$r$ $Equation of circumcircle is$

${(x - \frac{1}{2})}^{2} + {(y - \frac{3}{2})}^{2} = {(\sqrt{{(1 - 0)}^{2} + {(0 - 3)}^{2}} \div 2)}^{2} = \frac{5}{2} ...(i)$

$Required circle touches A B and A C, have radius r$

$∴$ $Equation be {(x - r)}^{2} + {(y - r)}^{2} = r^{2} ...(ii)$

$If circle in equation (ii) touches circumcircle internally, we have$ $d_{C_{1} C_{2}} =$ $| r_{1} - r_{2} |$

$\Rightarrow {(\frac{1}{2} - r)}^{2} + {(\frac{3}{2} - r)}^{2} = {(| \sqrt{\frac{5}{2}} - r |)}^{2}$

$\Rightarrow \frac{1}{4} + r^{2} - r + \frac{9}{4} + r^{2} - 3 r$

$\Rightarrow {(\sqrt{\frac{5}{2}} - r)}^{2} or {(r - \sqrt{\frac{5}{2}})}^{2}$

$\Rightarrow 2 r^{2} - 4 r + \frac{5}{2} = \frac{5}{2} + r^{2} - \sqrt{10} r$

$\Rightarrow r = 0 or 4 - \sqrt{10}$

$\Rightarrow r \approx 0.837 \approx 0.84$ (on rounding off)

Solution

Q.
Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]

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Solution

Q. Let ABC be the triangle with AB=1, AC=3 and ∠BAC=π2. If a circle of radius r>0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is _______. [2022]

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Q.
Let $A B C$ be the triangle with $A B = 1$ , $A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B$ , $A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _______. [2022]