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Question

In the given figure, $A B$ is the diameter of the circle, centered at $´ O to the power of ´ end exponent.$ If $angle C O A equals 60 degree comma blank A B equals 2 r comma blank A C equals d blank$ and $blank C D equals l comma$ then $l$ is equal to

$d \sqrt{3}$
$d / \sqrt{3}$
$3 d$
$\sqrt{3} d / 2$

The correct answer is: $d square root of 3$

$A C equals d comma blank O A equals O B equals r comma blank C D equals B D equals l comma blank angle C O A equals fraction numerator pi over denominator 3 end fraction$
$therefore A C to the power of 2 end exponent equals O A to the power of 2 end exponent plus O C to the power of 2 end exponent minus 2 blank A O blank O C cos invisible function application fraction numerator pi over denominator 3 end fraction$
$rightwards double arrow d to the power of 2 end exponent equals 2 r to the power of 2 end exponent minus 2 r to the power of 2 end exponent fraction numerator 1 over denominator 2 end fraction equals r to the power of 2 end exponent$

Also, $angle B O D equals angle C O D equals fraction numerator 2 pi over denominator 3 cross times 2 end fraction equals fraction numerator pi over denominator 3 end fraction blank rightwards double arrow tan invisible function application fraction numerator pi over denominator 3 end fraction equals fraction numerator B D over denominator O B end fraction equals fraction numerator l over denominator r end fraction rightwards double arrow l equals r square root of 3 equals d square root of 3$

Related Questions to study

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$I f tan invisible function application x equals n tan invisible function application y comma blank n element of R to the power of plus end exponent comma$ then the maximum value of $s e c to the power of 2 end exponent open parentheses x minus y close parentheses$ is equal to

If the equation $2 to the power of x end exponent plus 4 to the power of y end exponent equals 2 to the power of y end exponent plus 4 to the power of x end exponent$ is solved for $y$ in terms of $x comma blank$ where $x less than 0 comma$ then the sum of the solutions is

If the equation $2 to the power of x end exponent plus 4 to the power of y end exponent equals 2 to the power of y end exponent plus 4 to the power of x end exponent$ is solved for $y$ in terms of $x comma blank$ where $x less than 0 comma$ then the sum of the solutions is

$I f blank A$ and $blank B$ are acute positive angles satisfying the equations $3 sin to the power of 2 end exponent invisible function application A plus 2 sin to the power of 2 end exponent invisible function application B equals 1$ and $blank 3 sin invisible function application 2 A minus 2 sin invisible function application 2 B equals 0 comma$ then $A plus 2 blank B$ is equal to

$I f blank A$ and $blank B$ are acute positive angles satisfying the equations $3 sin to the power of 2 end exponent invisible function application A plus 2 sin to the power of 2 end exponent invisible function application B equals 1$ and $blank 3 sin invisible function application 2 A minus 2 sin invisible function application 2 B equals 0 comma$ then $A plus 2 blank B$ is equal to

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If $square root of log subscript 2 end subscript invisible function application x end root minus 0.5 equals log subscript 2 end subscript invisible function application square root of x comma$ then $x$ equals

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parallel

If both the distinct roots of the equation $open vertical bar sin invisible function application x close vertical bar to the power of 2 end exponent plus open vertical bar sin invisible function application x close vertical bar plus b equals 0 blank$ in $blank open square brackets 0 comma pi close square brackets$ are real, then the values of $b$ are

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If $open parentheses 4 close parentheses to the power of log subscript 9 end subscript invisible function application 3 end exponent plus open parentheses 9 close parentheses to the power of log subscript 2 end subscript invisible function application 4 end exponent equals open parentheses 10 close parentheses to the power of log subscript x end subscript invisible function application 83 end exponent comma$ then $x$ is equal to

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parallel

The value of $log invisible function application a b minus log invisible function application open vertical bar b close vertical bar equals$

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parallel

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If $log subscript 10 end subscript invisible function application open square brackets fraction numerator 1 over denominator 2 to the power of x end exponent plus blank x minus 1 end fraction close square brackets equals x open square brackets log subscript 10 end subscript invisible function application 5 minus 1 close square brackets comma blank t h e n blank x equals$

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r triangle $A B C comma blank R equals 5 divided by 2 blank$ and $blank r equals 1.$ Let $I$ be the incentre of the triangle and $D comma blank E$ and $blank F$ be the feet of the perpendiculars from $I blank$ to $blank B C comma blank C A blank$ and $blank A B comma$ respectively. The value of $fraction numerator I D cross times I E cross times I F over denominator I A cross times I B cross times I C end fraction blank$ is equal to

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parallel

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